Dirty Paper Channel Research Articles

Achieving the Dirty Paper Channel Capacity With Scalar Lattices and Probabilistic Shaping

Achieving the Dirty Paper Channel Capacity With Scalar Lattices and Probabilistic Shaping

Closed-Form Capacity Bounds for the Exponential Version of the Dirty Paper Channel

This letter presents closed-form capacity bounds for an exponential version of the dirty paper channel (EDPC). The EDPC can be used to model the non-coherent Gaussian dirty paper channel (GDPC). First, a superposition modulation-like technique is used to obtain a capacity upper bound for the EDPC. Then, a closed-form capacity lower bound for the EDPC is obtained by using a Costa-like strategy. Using this lower bound and exploiting the asymptotic properties of high signal to noise ratios (SNRs), we obtain the high-SNR capacity of the EDPC which is intriguingly analogous to the complex-valued GDPC. Finally, a closed-form capacity lower bound for the EDPC is given which is useful in the weak interference regime.

Proper multi‐layer coding in fading dirty‐paper channel

This study investigates multi-layer coding over a dirty-paper channel. First, it is demonstrated that superposition coding in such channel still achieves the capacity of interference-free additive white Gaussian noise channel when the transmitter is non-causally aware of interference signal. Then, the problem is extended to the dirty-paper block fading channel, where it is shown that in the lack of channel information at the transmitter, the so-called broadcast approach maximises the average achievable rate of such channel.

Lattice Codes Achieve the Capacity of Common Message Gaussian Broadcast Channels With Coded Side Information

Lattices possess elegant mathematical properties which have been previously used in the literature to show that structured codes can be efficient in a variety of communication scenarios, including coding for the additive white Gaussian noise channel, dirty-paper channel, Wyner–Ziv coding, coding for relay networks, and so forth. We consider the family of single-transmitter multiple-receiver Gaussian channels, where the source transmits a set of common messages to all the receivers (multicast scenario), and each receiver has coded side information , i.e., prior information in the form of linear combinations of the messages. This channel model is motivated by applications to multi-terminal networks, where the nodes may have access to coded versions of the messages from previous signal hops or through orthogonal channels. The capacity of this channel is known and follows from the work of Tuncel (2006), which is based on random coding arguments. In this paper, following the approach of Erez and Zamir, we design lattice codes for this family of channels when the source messages are symbols from a finite field $\mathbb {F}_{p}$ of prime size. Our coding scheme utilizes Construction A lattices designed over the same prime field $\mathbb {F}_{p}$ , and uses algebraic binning at the decoders to expurgate the channel code and obtain good lattice subcodes, for every possible set of linear combinations available as side information. The achievable rate of our coding scheme is a function of the size $p$ of underlying prime field, and approaches the capacity as $p$ tends to infinity.

On the Capacity of the Carbon Copy onto Dirty Paper Channel

The “carbon copy onto dirty paper” (CCDP) channel is the compound “writing on dirty paper” channel in which the channel output is obtained as the sum of the channel input, white Gaussian noise and a Gaussian state sequence randomly selected among a set possible realizations. The transmitter has non-causal knowledge of the set of possible state sequences but does not know which sequence is selected to produce the channel output. We study the capacity of the CCDP channel for two scenarios: 1) the state sequences are independent and identically distributed; and 2) the state sequences are scaled versions of the same sequence. In the first scenario, we show that a combination of superposition coding, time-sharing, and Gel’fand-Pinsker binning is sufficient to approach the capacity to within 3 bits per channel use for any number of possible state realizations. In the second scenario, we derive capacity to within 4 bits per channel use for the case of two possible state sequences. This result is extended to the CCDP channel with any number of possible state sequences under certain conditions on the scaling parameters, which we denote as “strong fading” regime. We conclude by providing some remarks on the capacity of the CCDP channel in which the state sequences have any jointly Gaussian distribution.

Ergodic Fading MIMO Dirty Paper and Broadcast Channels: Capacity Bounds and Lattice Strategies

A multiple-input multiple-output (MIMO) version of the dirty paper channel is studied, where the channel input and the dirt experience the same fading process and the fading channel state is known at the receiver (CSIR). This represents settings where signal and interference sources are co-located, such as in the broadcast channel. First, a variant of Costa's dirty paper coding (DPC) is presented, whose achievable rates are within a constant gap to capacity for all signal and dirt powers. Additionally, a lattice coding and decoding scheme is proposed, whose decision regions are independent of the channel realizations. Under Rayleigh fading, the gap to capacity of the lattice coding scheme vanishes with the number of receive antennas, even at finite Signal-to-Noise Ratio (SNR). Thus, although the capacity of the fading dirty paper channel remains unknown, this work shows it is not far from its dirt-free counterpart. The insights from the dirty paper channel directly lead to transmission strategies for the two-user MIMO broadcast channel (BC), where the transmitter emits a superposition of desired and undesired (dirt) signals with respect to each receiver. The performance of the lattice coding scheme is analyzed under different fading dynamics for the two users, showing that high-dimensional lattices achieve rates close to capacity.

Bounds on the Achievable Rates of Faded Dirty Paper Channel

Bounds on the Achievable Rates of Faded Dirty Paper Channel

Some “goodness” properties of LDA lattices

We study some structural properties of Construction-A lattices obtained from low density parity check (LDPC) codes over prime fields. Such lattices are called low density Construction-A (LDA) lattices, and permit low-complexity belief propagation decoding for transmission over Gaussian channels. It has been shown that LDA lattices achieve the capacity of the power constrained additive white Gaussian noise (AWGN) channel with closest lattice-point decoding, and simulations suggested that they perform well under belief propagation decoding. We continue this line of work, and prove that these lattices are good for packing and mean squared error (MSE) quantization, and that their duals are good for packing. With this, we can conclude that codes constructed using nested LDA lattices can achieve the capacity of the power constrained AWGN channel, the capacity of the dirty paper channel, the rates guaranteed by the compute-and-forward protocol, and the best known rates for bidirectional relaying with perfect secrecy.

Orthogonal space–time block coding over dirty paper channel: Outage capacity analysis

We analytically prove that the outage capacity of the multiple input multiple output (MIMO) which is affected by interference, that is non-causally available as well as the channel state information (CSI) for all users at the transmitter, has the free interference outage capacity. Specifically, Orthogonal space–time block coding (O-STBC) with two transmit antennas and arbitrary number of receive antennas is used for transmission and reception. We modify the random variable U which was proposed by Costa (1983) to account for the availability of the CSI at both the transmitter and receiver. Further, we use lattice dirty paper coding (DPC) to show that an interference free channel capacity can be achieved in the MIMO-OSTBC system. First, we derive the equivalent noise seen by the receiver using an equivalent lattice based dirty paper code. Then the optimal value of the power inflation factor is derived. Next, the channel capacity of the equivalent modulo lattice channel is computed. Finally, performance results in the case of various number of receive antennas are presented and showed that significant reduction in frame error probabilities can be obtained over a system that uses no interference cancellation.

Writing on Fading Paper, Dirty Tape With Little Ink: Wideband Limits for Causal Transmitter CSI

A wideband Rayleigh fading channel is considered with causal channel state information (CSI) at the transmitter and no receiver CSI. A simple orthogonal code with energy detection rule at the receiver (similar to pulse position modulation in IEEE Trans. Inf. Theory, vol. 46, no. 4, Apr. 2000 and IEEE Trans. Inf. Theory, vol. 52 no. 5, May 2006) is shown to achieve the capacity of this channel in the wideband limit. This strategy transmits energy only when the channel gain exceeds a threshold, hence only needs causal transmitter CSI. In the wideband limit, this capacity without any receiver CSI is the same as the capacity with full receiver CSI, which is proportional to the logarithm of the bandwidth. Similar threshold-based pulse position modulation is shown to achieve the capacity per unit cost of the dirty-tape channel (dirty paper channel with causal transmitter CSI and no receiver CSI), which equals its capacity per unit cost with full receiver CSI. Then, a general discrete channel with i.i.d. states is considered. Each input has an associated cost and a zero cost input “0” exists. The channel state is assumed to be known at the transmitter in a causal manner. Capacity per unit cost is found for this channel and a simple orthogonal code is shown to achieve this capacity. Later, a novel orthogonal coding scheme is proposed for the case of causal transmitter CSI and a condition for equivalence of capacity per unit cost for causal and noncausal transmitter CSI is derived.

An Upper Bound on the Capacity of the Causal Dirty-Paper Channel

A bound on the capacity of the causal dirty-paper channel with arbitrary interference and general independent identically distributed additive noise is derived. In particular, it is shown that for the case of Gaussian noise, the capacity is upper bounded by log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (1+SNR/ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</i> ) bits per real dimension. This bound is useful for SNR ≤ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</i> -2) .

Lattice Strategies for the Dirty Multiple Access Channel

In Costa's dirty-paper channel, Gaussian random binning is able to eliminate the effect of interference which is known at the transmitter, and thus achieve capacity. We examine a generalization of the dirty-paper problem to a multiple access channel (MAC) setup, where structured (lattice-based) binning seems to be necessary to achieve capacity. In the dirty-MAC, two additive interference signals are present, one known to each transmitter but none to the receiver. The achievable rates using Costa's Gaussian binning vanish if both interference signals are strong. In contrast, it is shown that lattice-strategies (“lattice precoding”) can achieve positive rates, independent of the interference power. Furthermore, in some cases-which depend on the noise variance and power constraints-high-dimensional lattice strategies are in fact optimal. In particular, they are optimal in the limit of high SNR-where the capacity region of the dirty MAC with strong interference approaches that of a clean MAC whose power is governed by the minimum of the users' powers rather than their sum. The rate gap at high SNR between lattice-strategies and optimum (rather than Gaussian) random binning is conjectured to be 1/2 log2(πe/6) ≈ 0.254 bit. Thus, the doubly dirty MAC is another instance of a network setting, like the Korner-Marton problem, where (linear) structured coding is potentially better than random binning.

Analog Matching of Colored Sources to Colored Channels

Analog (uncoded) transmission provides a simple and robust scheme for communicating a Gaussian source over a Gaussian channel under the mean-squared-error (MSE) distortion measure. Unfortunately, its performance is usually inferior to the all-digital, separation-based source-channel coding solution, which requires exact knowledge of the channel at the encoder. The loss comes from the fact that except for very special cases, e.g., white source and channel of matching bandwidth (BW), it is impossible to achieve perfect matching of source to channel and channel to source by linear means. We show that by combining prediction and modulo-lattice operations, it is possible to match any colored Gaussian source to any colored Gaussian noise channel (of possibly different BW), hence achieve Shannon's optimum attainable performance R(D)=C. Furthermore, when the source and channel BWs are equal (but otherwise their spectra are arbitrary), this scheme is asymptotically robust in the sense that for high signal-to-noise ratio (SNR) a single encoder (independent of the noise variance) achieves the optimum performance. The derivation is based upon a recent modulo-lattice modulation scheme for transmitting a Wyner-Ziv source over a dirty-paper channel.

Secret-Key Agreement With Channel State Information at the Transmitter

We study the capacity of secret-key agreement over a wiretap channel with state parameters. The transmitter communicates to the legitimate receiver and the eavesdropper over a discrete memoryless wiretap channel with a memoryless state sequence. The transmitter and the legitimate receiver generate a shared secret key, that remains secret from the eavesdropper. No public discussion channel is available. The state sequence is known noncausally to the transmitter. We derive lower and upper bounds on the secret-key capacity. The lower bound involves constructing a common state reconstruction sequence at the legitimate terminals and binning the set of reconstruction sequences to obtain the secret-key. For the special case of Gaussian channels with additive interference (secret-keys from dirty paper channel) our bounds differ by 0.5 bit/symbol and coincide in the high signal-to-noise-ratio and high interference-to-noise-ratio regimes. For the case when the legitimate receiver is also revealed the state sequence, we establish that our lower bound achieves the the secret-key capacity. In addition, for this special case, we also propose another scheme that attains the capacity and requires only causal side information at the transmitter and the receiver.

On the robustness of dirty paper coding

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

Bounds and lattice-based transmission strategies for the phase-faded dirty-paper channel

We consider a fading version of the dirty-paper problem, as proposed by Grover and Sahai. In this formulation, the various signals involved are complex-valued, and the interference (known only to the transmitter) is multiplied by a random complex-valued coefficient, whose phase is known only to the receiver. We focus on a compound channel formulation, and seek to maximize the worst-case performance. We present an achievable strategy modeled on the lattice-based approach of Erez, Shamai and Zamir and propose heuristic methods to optimize its parameters. We also derive an upper bound on the maximum achievable transmission rates. Our bounds are shown to be tight in some settings, yielding a complete characterization of capacity. We also provide simulation results, indicating the practical effectiveness of our approaches.

On rate and BER analysis for finite-dimensional lattice coding for the dirty paper channel

Costa's dirty paper coding (DPC) offers a good framework for precoding for transmission over additive Gaussian channels with additive states non-causally known at the transmitter but not at the receiver. In this paper, rate calculation, error probability analysis and code design for DPC are investigated from a practical point-of-view. Based on Monte-Carlo numerical integration and simulations, we first show that the gap to the full AWGN capacity can be partially bridged using some finite-dimensional lattices with good packing and quantizing properties. Then the difficult problem of codebook selection is addressed through some illustrative examples. Analysis sheds light on the dual roles of “packing” and “shaping” as well as on their inter-connection at finite-dimensional coding, by opposition to the asymptotic case (i.e., infinite dimension) where the two coding components are decoupled.

Sensing in Cognitive Radio Channels: A Theoretical Perspective

The cognitive radio paradigm is based on the ability of sensing the radio environment in order to make informed decisions. This paper describes the effects of sensing on the cognitive radio channels capacity region. Sensing is modeled as a compression channel, which results in partial knowledge of the primary messages at the cognitive transmitter. This model enables to impose constraints on the sensing strategy. First, the dirty paper channel capacity is derived when the channel encoder knows partially the side information. Then, the capacity area of the Gaussian cognitive channel with partial information is derived. Finally, numerical results illustrate the capacity reduction associated with constrained sensing, in comparison to the capacity of the cognitive radio channel.

Multiple Antenna Broadcast Channels With Shape Feedback and Limited Feedback

In this paper, we consider two different models of partial channel state information at the base station transmitter (CSIT) for multiple antenna broadcast channels: 1) the shape feedback model where the normalized channel vector of each user is available at the base station and 2) the limited feedback model where each user quantizes its channel vector according to a rotated codebook that is optimal in the sense of mean squared error and feeds back the codeword index. This paper is focused on characterizing the sum rate performance of both zero-forcing dirty paper coding (ZFDPC) systems and channel inversion (CI) systems under the given two partial CSIT models. Intuitively speaking, a system with shape feedback loses the sum rate gain of adaptive power allocation. However, shape feedback still provides enough channel knowledge for ZFDPC and CI to approach their own optimal throughput in the high signal-to-noise ratio (SNR) regime. As for limited feedback, we derive sum rate bounds for both signaling schemes and link their throughput performance to some basic properties of the quantization codebook. Interestingly, we find that limited feedback employing a fixed codebook leads to a sum rate ceiling for both schemes for asymptotically high SNR.

Information Rates Subject to State Masking

We consider the problem of rate-R, channel coding with causal/noncausal side information at the transmitter, under an additional requirement of minimizing the amount of information that can be learned from the channel output about the state sequence, which is defined in terms of the mutual information between the state sequence and the channel output sequence. A single-letter characterization is provided for the achievable region of pairs {(R, E)}. Explicit results for the Gaussian case (Costa's dirty-paper channel) are derived in full detail.