We present a deterministic channel model which captures several key features of multiuser wireless communication. We consider a model for a wireless network with nodes connected by such deterministic channels, and present an exact characterization of the end-to-end capacity when there is a single source and a single destination and an arbitrary number of relay nodes. This result is a natural generalization of the max-flow min-cut theorem for wireline networks. Finally to demonstrate the connections between deterministic model and Gaussian model, we look at two examples: the single-relay channel and the diamond network. We show that in each of these two examples, the capacity-achieving scheme in the corresponding deterministic model naturally suggests a scheme in the Gaussian model that is within 1 bit and 2 bit respectively from cut-set upper bound, for all values of the channel gains. This is the first part of a two-part paper; the sequel [1] will focus on the proof of the max-flow min-cut theorem of a class of deterministic networks of which our model is a special case.
Finite field addition makes the model much more tractable, and neglecting the 1-bit carryover from one level to the next introduce a small error when the SNR is high. Other works @cite_7 have also exploited the simplicity of finite-field addition over real addition. Aref @cite_1 is one of the earliest works that use deterministic models for relay networks, and for which he proved a capacity result for the single-source-single-destination case. However, his model only captures the broadcast aspect but not the superposition aspect. This work was later extended to the multicast setting by Ratnaker and Kramer @cite_8 . Aref and El Gamal @cite_0 also computed the capacity of the semi-determinstic relay channel but only with a single relay. @cite_4 also uses finite-field deterministic addition to model the superposition property, but they do not have the notion of signal scale and the channel as sending some of the signal scales to below noise level. Instead they use random erasures to model noise.
@cite_4 @cite_7 @cite_8 @cite_1 @cite_0
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We consider a finite-field model for the wireless broadcast and additive interference network (WBAIN), both in the presence and absence of fading. We show that the single-source unicast capacity (with extension to multicast) of a WBAIN with or without fading can be upper bounded by the capacity of an equivalent broadcast erasure network. We further present a coding strategy for WBAINs with i.i.d. and uniform fading based on random linear coding at each node that achieves a rate differing from the upper bound by no more than O(1 q), where q is the field size. Using these results, we show that channel fading in conjunction with network coding can lead to large gains in the unicast (multicast) capacity as compared to no fading The multicast capacity is determined for networks that have deterministic channels with broadcasting at the transmitters and no interference at the receivers. The multicast capacity is shown to have a cut-set interpretation. It is further shown that one cannot always layer channel and network coding in such networks. The proof of the latter result partially generalizes to discrete memoryless broadcast channels and is used to bound the common rate for problems where one achieves a cut bound on throughput. A centrifugating device for biological liquids, e.g. blood, in which a rotatable container carries a specially shaped seal that surrounds and bears on a fixed assembly with a minimum area of interface between the fixed and rotating parts. This seal is disposed outside the path of the liquid to be treated. The fixed assembly, in turn, is releasably carried by a bracket, the bracket being selectively longitudinally extensible as well as selectively adjustably swingable about a vertical axis of oscillation eccentric to the centrifuge, thereby to permit exact positioning of the fixed assembly coaxially of the rotatable container. The parts are so simple and inexpensive in construction that at least some of them can be used once and thrown away. Moreover, the fixed assembly is easily insertable in sealed relationship in any of a variety of containers, by the simplest of manual assembly and disassembly operations. The capacity of the class of relay channels with sender x_ 1 , a relay sender x_ 2 , a relay receiver y_ 1 =f(x_ 1 ,x_ 2 ) , and ultimate receiver y is proved to be C = p(x_ 1 ,x_ 2 ) I(X_ 1 , X_ 2 ; Y), H(Y_ 1 |X_ 2 )+I(X_ 1 ;Y|X_ 2 ,Y_ 1 ) .