Outage probabilities and single-hop throughput are two important performance metrics that have been evaluated for certain specific types of wireless networks. However, there is a lack of comprehensive results for larger classes of networks, and there is no systematic approach that permits the convenient comparison of the performance of networks with different geometries and levels of randomness. The uncertainty cube is introduced to categorize the uncertainty present in a network. The three axes of the cube represent the three main potential sources of uncertainty in interference-limited networks: the node distribution, the channel gains (fading), and the channel access scheme (set of transmitting nodes). For the performance analysis, a new parameter, the so- called spatial contention, is defined. It measures the slope of the outage probability in an ALOHA network as a function of the transmit probability p at p = 0. Outage is defined as the event that the signal-to-interference ratio (SIR) is below a certain threshold in a given time slot. It is shown that the spatial contention is sufficient to characterize outage and throughput in large classes of wireless networks, corresponding to different positions on the uncertainty cube. Existing results are placed in this framework, and new ones are derived. Further, interpreting the outage probability as the SIR distribution, the ergodic capacity of unit-distance links is determined and compared to the throughput achievable for fixed (yet optimized) transmission rates.
In @cite_13 @cite_1 , was calculated for a two-dimensional random network with Rayleigh fading and ALOHA. Ignoring the noise, they obtained (see [Eqn. (3.4)] net:Baccelli06 , [(Eqn. (A.11)] net:Zorzi95 ) with The subscript 2 in @math indicates that this is a constant for the two-dimensional case. Useful values include @math and @math . @math , so @math as @math for any @math . The spatial contention is @math .
@cite_1 @cite_13
2137891392 2018778093 2143664144 2542299727
Spatial Aloha is probably the simplest medium access protocol to be used in a large mobile ad hoc network: each station tosses a coin independently of everything else and accesses the channel if it gets heads. In a network where stations are randomly and homogeneously located in the Euclidean plane, there is a way to tune the bias of the coin so as to obtain the best possible compromise between spatial reuse and per transmitter throughput. This paper shows how to address this questions using stochastic geometry and more precisely Poisson shot noise field theory. The theory that is developed is fully computational and leads to new closed form expressions for various kinds of spatial averages (like e.g. outage, throughput or transport). It also allows one to derive general scaling laws that hold for general fading assumptions. We exemplify its flexibility by analyzing a natural variant of Spatial Aloha that we call Opportunistic Aloha and that consists in replacing the coin tossing by an evaluation of the quality of the channel of each station to its receiver and a selection of the stations with good channels (e.g. fading) conditions. We show how to adapt the general machinery to this variant and how to optimize and implement it. We show that when properly tuned, Opportunistic Aloha very significantly outperforms Spatial Aloha, with e.g. a mean throughput per unit area twice higher for Rayleigh fading scenarios with typical parameters. Consider the problem of recovering a measure @math supported on a lattice of span @math , when measurements are only available concerning the Fourier Transform @math at frequencies @math . If @math is much smaller than the Nyquist frequency @math and the measurements are noisy, then, in general, stable recovery of @math is impossible. In this paper it is shown that if, in addition, we know that the measure @math satisfies certain sparsity constraints, then stable recovery is possible. Say that a set has Rayleigh index less than or equal to R if in any interval of length @math there are at most R elements. Indeed, if the (unknown) support of @math is known, a priori, to have Rayleigh index at most R, then stable recovery is possible with a stability coefficient that grows at most like @math as @math . This result validates certain practical efforts, in spectroscopy, seismic prospecting, and astrono... We consider the generalized dirty-paper channel Y=X+S+N,E X sup 2 spl les P sub X , where N is not necessarily Gaussian, and the interference S is known causally or noncausally to the transmitter. We derive worst case capacity formulas and strategies for "strong" or arbitrarily varying interference. In the causal side information (SI) case, we develop a capacity formula based on minimum noise entropy strategies. We then show that strategies associated with entropy-constrained quantizers provide lower and upper bounds on the capacity. At high signal-to-noise ratio (SNR) conditions, i.e., if N is weak relative to the power constraint P sub X , these bounds coincide, the optimum strategies take the form of scalar lattice quantizers, and the capacity loss due to not having S at the receiver is shown to be exactly the "shaping gain" 1 2log(2 spl pi e 12) spl ap 0.254 bit. We extend the schemes to obtain achievable rates at any SNR and to noncausal SI, by incorporating minimum mean-squared error (MMSE) scaling, and by using k-dimensional lattices. For Gaussian N, the capacity loss of this scheme is upper-bounded by 1 2log2 spl pi eG( spl Lambda ), where G( spl Lambda ) is the normalized second moment of the lattice. With a proper choice of lattice, the loss goes to zero as the dimension k goes to infinity, in agreement with the results of Costa. These results provide an information-theoretic framework for the study of common communication problems such as precoding for intersymbol interference (ISI) channels and broadcast channels. We investigate recovery of nonnegative vectors from nonadaptive compressive measurements in the presence of noise of unknown power. It is known in the literature that under additional assumptions on the measurement design recovery of such vectors is possible with nonnegative least squares without any regularization. We show that uniqueness results known for the noiseless case carry over to robust guarantees in the noisy setting. We present guarantees which hold instantaneously by connecting the relation to the robust nullspace property. As an important example, we prove that an m × n random iid. 0 1-valued Bernoulli matrix with m = O(s log(n)) rows admits the robust nullspace property with high probability and meets the design requirements for nonnegative least squares recovery. Our analysis is motivated by applications in wireless network activity detection.