103,858 research outputs found
A digital interface for Gaussian relay networks: lifting codes from the discrete superposition model to Gaussian relay networks
For every Gaussian relay network with a single source-destination pair, it is
known that there exists a corresponding deterministic network called the
discrete superposition network that approximates its capacity uniformly over
all SNR's to within a bounded number of bits. The next step in this program of
rigorous approximation is to determine whether coding schemes for discrete
superposition models can be lifted to Gaussian relay networks with a bounded
rate loss independent of SNR. We establish precisely this property and show
that the superposition model can thus serve as a strong surrogate for designing
codes for Gaussian relay networks.
We show that a code for a Gaussian relay network, with a single
source-destination pair and multiple relay nodes, can be designed from any code
for the corresponding discrete superposition network simply by pruning it. In
comparison to the rate of the discrete superposition network's code, the rate
of the Gaussian network's code only reduces at most by a constant that is a
function only of the number of nodes in the network and independent of channel
gains.
This result is also applicable for coding schemes for MIMO Gaussian relay
networks, with the reduction depending additionally on the number of antennas.
Hence, the discrete superposition model can serve as a digital interface for
operating Gaussian relay networks.Comment: 5 pages, 2010 IEEE Information Theory Workshop, Cair
A digital interface for Gaussian relay and interference networks: Lifting codes from the discrete superposition model
For every Gaussian network, there exists a corresponding deterministic
network called the discrete superposition network. We show that this discrete
superposition network provides a near-optimal digital interface for operating a
class consisting of many Gaussian networks in the sense that any code for the
discrete superposition network can be naturally lifted to a corresponding code
for the Gaussian network, while achieving a rate that is no more than a
constant number of bits lesser than the rate it achieves for the discrete
superposition network. This constant depends only on the number of nodes in the
network and not on the channel gains or SNR. Moreover the capacities of the two
networks are within a constant of each other, again independent of channel
gains and SNR. We show that the class of Gaussian networks for which this
interface property holds includes relay networks with a single
source-destination pair, interference networks, multicast networks, and the
counterparts of these networks with multiple transmit and receive antennas.
The code for the Gaussian relay network can be obtained from any code for the
discrete superposition network simply by pruning it. This lifting scheme
establishes that the superposition model can indeed potentially serve as a
strong surrogate for designing codes for Gaussian relay networks.
We present similar results for the K x K Gaussian interference network, MIMO
Gaussian interference networks, MIMO Gaussian relay networks, and multicast
networks, with the constant gap depending additionally on the number of
antennas in case of MIMO networks.Comment: Final versio
Liouville Numbers and Schanuel's Conjecture
In this paper, using an argument of P. Erdos, K. Alniacik and E. Saias, we
extend earlier results on Liouville numbers, due to P. Erdos, G.J. Rieger, W.
Schwarz, K. Alniacik, E. Saias, E.B. Burger. We also produce new results of
algebraic independence related with Liouville numbers and Schanuel's
Conjecture, in the framework of G delta-subsets.Comment: Archiv der Math., to appea
Liouville numbers, Liouville sets and Liouville fields
Following earlier work by E.Maillet 100 years ago, we introduce the
definition of a Liouville set, which extends the definition of a Liouville
number. We also define a Liouville field, which is a field generated by a
Liouville set. Any Liouville number belongs to a Liouville set S having the
power of continuum and such that the union of S with the rational number field
is a Liouville field.Comment: Proceedings of the American Mathematical Society, to appea
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