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Allerton Conference on Communication, Control, and Computing

Optimal jamming Over additive noise: Vector source-channel case

Emrah Akyol, Kenneth H. Rose

Published Oct 1, 2013• 3 citations
Publisher

Abstract

This paper extends the recent analysis of optimal zero-delay jamming problem, specialized to scalar sources over scalar additive noise channels, to vector spaces. Particularly, we find the saddle point solution to the jamming problem in “vector spaces”, within its most general, non-Gaussian setting; the problem has been open even for the important special case of Gaussian source and channel. Similar to the scalar setting, linearity conditions for encoding and decoding mappings play a pivotal role in jamming, in the sense that the optimal jamming strategy is to effectively force both transmitter and receiver to default to linear mappings, i.e., the jammer ensures, whenever possible, that the transmitter and receiver cannot benefit from non-linear strategies. The optimal strategy is then “randomized” linear encoding for transmitter and generating independent noise for the jamming. Moreover, the optimal jammer allocates power according to the well-known “water-filling” over the eigenvalues of the channel noise, and the density of the jamming noise that render the optimal encoding and decoding mappings linear, is determined by the power constraints and the source density.

DOI
10.1109/allerton.2013.6736680

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This paper extends the recent analysis of optimal zero-delay jamming problem, specialized to scalar sources over scalar additive noise channels, to vector spaces. Particularly, we find the saddle point solution to the jamming problem in “vector spaces”, within its most general, non-Gaussian setting; the problem has been open even for the important special case of Gaussian source and channel. Similar to the scalar setting, linearity conditions for encoding and decoding mappings play a pivotal role in jamming, in the sense that the optimal jamming strategy is to effectively force both transmitter and receiver to default to linear mappings, i.e., the jammer ensures, whenever possible, that the transmitter and receiver cannot benefit from non-linear strategies. The optimal strategy is then “randomized” linear encoding for transmitter and generating independent noise for the jamming. Moreover, the optimal jammer allocates power according to the well-known “water-filling” over the eigenvalues of the channel noise, and the density of the jamming noise that render the optimal encoding and decoding mappings linear, is determined by the power constraints and the source density.
This paper considers the problem of optimal zero-delay jamming over an additive noise channel. Building on a sequence of recent results on conditions for linearity of optimal estimation, and of optimal mappings in source-channel coding, the saddle-point solution to the jamming problem is derived for general sources and channels, without recourse to Gaussianity assumptions. The linearity conditions are shown to play a pivotal role in jamming, in the sense that the optimal jamming strategy is to effectively force both the transmitter and the receiver to default to linear mappings, i.e., the jammer ensures, whenever possible, that the transmitter and the receiver cannot benefit from non-linear strategies. This result is shown to subsume the known result for Gaussian source and channel. The conditions and general settings where such unbeatable strategy can indeed be achieved by the jammer are analyzed. Moreover, a numerical procedure is provided to approximate the optimal jamming strategy in the remaining (source-channel) cases where the jammer cannot impose linearity on the transmitter and the receiver. Next, the analysis is extended to vector sources and channels. This extension involves a new aspect of optimization: the allocation of available transmit and jamming power over source and channel components. Similar to the scalar setting, the saddle-point solution is derived using the linearity conditions in vector spaces. The optimal power allocation strategies for the jammer and the transmitter have an intuitive interpretation as the jammer allocates power according to water-filling over the channel eigenvalues, while the transmitter performs water-pouring (reverse water-filling) over the source eigenvalues.
This paper considers the problem of optimal zero-delay jamming over an additive noise channel. Early work had already solved this problem for a Gaussian source and channel. Building on a sequence of recent results on conditions for linearity of optimal estimation, and of optimal mappings in source-channel coding, we derive the saddle-point solution to the jamming problem for general sources and channels, without recourse to Gaussian assumptions. We show that linearity conditions play a pivotal role in jamming, in the sense that the optimal jamming strategy is to effectively force both transmitter and receiver to default to linear mappings, i.e., the jammer ensures, whenever possible, that the transmitter and receiver cannot benefit from non-linear strategies. This result is shown to subsume the known result for Gaussian source and channel. We analyze conditions and general settings where such unbeatable strategy can indeed be achieved by the jammer. Moreover, we provide the procedure to approximate optimal jamming in the remaining (source-channel) cases where the jammer cannot impose linearity on the transmitter and the receiver.
This paper considers the problem of optimal zero-delay jamming over an additive noise channel. Early work had solved this problem for a Gaussian source and a Gaussian channel. Building on a sequence of recent results on conditions for linearity of optimal estimation, and of optimal mappings in source-channel coding, we derive the saddle-point solution to the jamming problem for general sources and channels, without recourse to Gaussianness assumptions. We show that linearity conditions play a pivotal role in jamming, in the sense that the optimal jamming strategy is to effectively force both the transmitter and the receiver to default to linear mappings, i.e., the jammer ensures, whenever possible, that the transmitter and the receiver cannot benefit from non-linear strategies. This result is shown to subsume the known result for Gaussian source and channel. We analyze conditions and general settings where such unbeatable strategy can indeed be achieved by the jammer. Moreover, we provide a procedure to approximate optimal jamming in the remaining (source-channel) cases where the jammer cannot impose linearity on the transmitter and the receiver.
The problem of transmitting a sequence of identically distributed independent Gaussian random variables through a Gaussian memoryless channel with a given input power constraint, in the presence of an intelligent jammer, is considered. The jammer taps the channel and feeds back a signal, at a given energy level, for the purpose of jamming the transmitting sequence. Under a square-difference distortion measure which is to be maximized by the jammer and to be minimized by the transmitter and the receiver, this correspondence obtains the complete set of optimal (saddle-point) policies. The solution is essentially unique, and it is structurally different in three different regions in the parameter space, which are determined by the signal-to-noise ratios and relative magnitudes of the noise variances. The best (maximin) policy of the jammer is either to choose a linear function of the measurement he receives through channel-tapping, or to choose, in addition (and additively), an independent Gaussian noise sequence, depending on the region where the parameters lie. The optimal (minimax) policy of the transmitter is to amplify the input sequence to the given power level by a linear transformation, and that of the receiver is to use a Bayes estimator.
Worst-case jamming of legitimate communications over multiple-input multiple-output Gaussian channels is studied in this paper. A worst-case scenario with a `smart' jammer that knows all channels and the transmitter's strategy and is only power limited is considered. It is shown that the simplification of the system model by neglecting the properties of the jamming channel leads to a loss of important insights regarding the effects of the jamming power and jamming channel on optimal jamming strategies of the jammer. Without neglecting the jamming channel, a lower-bound on the rate of legitimate communication subject to jamming is derived, and conditions for this bound to be positive are given. The lower-bound rate can be achieved regardless of the quality of the jamming channel, the power limit of the jammer, and the transmit strategy of the jammer. Moreover, general forms of an optimal jamming strategy, on the basis of which insights into the effect of jamming power and jamming channel are exposed, are given. It is shown that the general forms can lead to closed-form optimal jamming solutions when the power limit of the jammer is larger than a threshold. Subsequently, the scenario in which the effect of jamming dominates the effect of noise (the case of practical interest) is considered, and an optimal jamming strategy is derived in closed-form. Simulation examples demonstrate lower-bound rates, performance of the derived jamming strategy, and an effect of inaccurate channel information on the jamming strategy.
Abstract : The additive infinite-dimensional Gaussian channel subject to jamming is modeled as a two-person zero-sum game with mutual information as the payoff function. The jammer's noise is added to the ambient Gaussian noise. The coder's signal energy is subject to a constraint is necessary in order that the capacity without feedback be finite. It is shown that use of this same RKHS constraint on the jammer's process is too strong; the jammer would then not be able to reduce capacity, regardless of the amount of jamming energy available. The constraint on the jammer is thus on the total jamming energy, without regard to its distribution relative to that of the ambient noise energy. The existence of a saddle value for the problem does not follow from the von Neuman minimax theorem in the original problem formulation. However, a solution is shown to exist. A saddle point, saddle value, and the jammer's minimax strategy are determined. The solution is a function of the problem parameters: the constraint on the coder, the constraint on the jammer, and the covariance of the ambient Gaussian noise.