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Register a new userRateless Coding for Gaussian Channels
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Added by
Gregory Wornell
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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A rateless code-i.e., a rate-compatible family of codes-has the property that codewords of the higher rate codes are prefixes of those of the lower rate ones. A perfect family of such codes is one in which each of the codes in the family is capacity-achieving. We show by construction that perfect rateless codes with low-complexity decoding algorithms exist for additive white Gaussian noise channels. Our construction involves the use of layered encoding and successive decoding, together with repetition using time-varying layer weights. As an illustration of our framework, we design a practical three-rate code family. We further construct rich sets of near-perfect rateless codes within our architecture that require either significantly fewer layers or lower complexity than their perfect counterparts. Variations of the basic construction are also developed, including one for time-varying channels in which there is no a priori stochastic model.
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In this paper the performance limits and design principles of rateless codes over fading channels are studied. The diversity-multiplexing tradeoff (DMT) is used to analyze the system performance for all possible transmission rates. It is revealed from the analysis that the design of such rateless codes follows the design principle of approximately universal codes for parallel multiple-input multiple-output (MIMO) channels, in which each sub-channel is a MIMO channel. More specifically, it is shown that for a single-input single-output (SISO) channel, the previously developed permutation codes of unit length for parallel channels having rate LR can be transformed directly into rateless codes of length L having multiple rate levels (R, 2R, . . ., LR), to achieve the DMT performance limit.
A rateless transmission architecture is developed for communication over Gaussian intersymbol interference channels, based on the concept of super-Nyquist (SNQ) signaling. In such systems, the signaling rate is chosen significantly higher than the Nyquist rate of the system. We show that such signaling, when used in conjunction with good off-the-shelf base codes, simple linear redundancy, and minimum mean-square error decision feedback equalization, results in capacity-approaching, low-complexity rateless codes for the time-varying intersymbol-interference channel. Constructions for both single-input / single-output (SISO) and multi-input / multi-output (MIMO) ISI channels are developed.
The paper studies a class of three user Gaussian interference channels. A new layered lattice coding scheme is introduced as a transmission strategy. The use of lattice codes allows for an alignment of the interference observed at each receiver. The layered lattice coding is shown to achieve more than one degree of freedom for a class of interference channels and also achieves rates which are better than the rates obtained using the Han-Kobayashi coding scheme.
Non-malleable codes protect against an adversary who can tamper with the coded message by using a tampering function in a specified function family, guaranteeing that the tampering result will only depend on the chosen function and not the coded message. The codes have been motivated for providing protection against tampering with hardware that stores the secret cryptographic keys, and have found significant attention in cryptography. Traditional Shannon model of communication systems assumes the communication channel is perfectly known to the transmitter and the receiver. Arbitrary Varying Channels (AVCs) remove this assumption and have been used to model adversarially controlled channels. Transmission over these channels has been originally studied with the goal of recovering the sent message, and more recently with the goal of detecting tampering with the sent messages. In this paper we introduce non-malleability as the protection goal of message transmission over these channels, and study binary (discrete memoryless) AVCs where possible tampering is modelled by the set of channel states. Our main result is that non-malleability for these channels is achievable at a rate asymptotically approaching 1. We also consider the setting of an AVC with a special state s*, and the additional requirement that the message must be recoverable if s* is applied to all the transmitted bits. We give the outline of a message encoding scheme that in addition to non-malleability, can provide recovery for all s* channel.
In cite{butman1976} the linear coding scheme is applied, $X_t =g_tBig(Theta - {bf E}Big{ThetaBig|Y^{t-1}, V_0=v_0Big}Big)$, $t=2,ldots,n$, $X_1=g_1Theta$, with $Theta: Omega to {mathbb R}$, a Gaussian random variable, to derive a lower bound on the feedback rate, for additive Gaussian noise (AGN) channels, $Y_t=X_t+V_t, t=1, ldots, n$, where $V_t$ is a Gaussian autoregressive (AR) noise, and $kappa in [0,infty)$ is the total transmitter power. For the unit memory AR noise, with parameters $(c, K_W)$, where $cin [-1,1]$ is the pole and $K_W$ is the variance of the Gaussian noise, the lower bound is $C^{L,B} =frac{1}{2} log chi^2$, where $chi =lim_{nlongrightarrow infty} chi_n$ is the positive root of $chi^2=1+Big(1+ frac{|c|}{chi}Big)^2 frac{kappa}{K_W}$, and the sequence $chi_n triangleq Big|frac{g_n}{g_{n-1}}Big|, n=2, 3, ldots,$ satisfies a certain recursion, and conjectured that $C^{L,B}$ is the feedback capacity. In this correspondence, it is observed that the nontrivial lower bound $C^{L,B}=frac{1}{2} log chi^2$ such that $chi >1$, necessarily implies the scaling coefficients of the feedback code, $g_n$, $n=1,2, ldots$, grow unbounded, in the sense that, $lim_{nlongrightarrowinfty}|g_n| =+infty$. The unbounded behaviour of $g_n$ follows from the ratio limit theorem of a sequence of real numbers, and it is verified by simulations. It is then concluded that such linear codes are not practical, and fragile with respect to a mismatch between the statistics of the mathematical model of the channel and the real statistics of the channel. In particular, if the error is perturbed by $epsilon_n>0$ no matter how small, then $X_n =g_tBig(Theta - {bf E}Big{ThetaBig|Y^{t-1}, V_0=v_0Big}Big)+g_n epsilon_n$, and $|g_n|epsilon_n longrightarrow infty$, as $n longrightarrow infty$.
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