Rate-Constrained Wireless Networks with Fading Channels: Interference-Limited and Noise-Limited Regimes

A network of $n$ wireless communication links is considered in a Rayleigh fading environment. It is assumed that each link can be active and transmit with a constant power $P$ or remain silent. The objective is to maximize the number of active links such that each active link can transmit with a constant rate $\lambda$. An upper bound is derived that shows the number of active links scales at most like $\frac{1}{\lambda} \log n$. To obtain a lower bound, a decentralized link activation strategy is described and analyzed. It is shown that for small values of $\lambda$, the number of supported links by this strategy meets the upper bound; however, as $\lambda$ grows, this number becomes far below the upper bound. To shrink the gap between the upper bound and the achievability result, a modified link activation strategy is proposed and analyzed based on some results from random graph theory. It is shown that this modified strategy performs very close to the optimum. Specifically, this strategy is \emph{asymptotically almost surely} optimum when $\lambda$ approaches $\infty$ or 0. It turns out the optimality results are obtained in an interference-limited regime. It is demonstrated that, by proper selection of the algorithm parameters, the proposed scheme also allows the network to operate in a noise-limited regime in which the transmission rates can be adjusted by the transmission powers. The price for this flexibility is a decrease in the throughput scaling law by a multiplicative factor of $\log \log n$.Comment: Submitted to IEEE Trans. Information Theor

Bandwidth Partitioning in Decentralized Wireless Networks

This paper addresses the following question, which is of interest in the design of a multiuser decentralized network. Given a total system bandwidth of W Hz and a fixed data rate constraint of R bps for each transmission, how many frequency slots N of size W/N should the band be partitioned into in order to maximize the number of simultaneous links in the network? Dividing the available spectrum results in two competing effects. On the positive side, a larger N allows for more parallel, noninterfering communications to take place in the same area. On the negative side, a larger N increases the SINR requirement for each link because the same information rate must be achieved over less bandwidth. Exploring this tradeoff and determining the optimum value of N in terms of the system parameters is the focus of the paper. Using stochastic geometry, the optimal SINR threshold - which directly corresponds to the optimal spectral efficiency - is derived for both the low SNR (power-limited) and high SNR (interference-limited) regimes. This leads to the optimum choice of the number of frequency bands N in terms of the path loss exponent, power and noise spectral density, desired rate, and total bandwidth.Comment: Revised for IEEE Trans. Wireless Communications, April 2008 (initially submitted Nov. 2007). Results shown to apply to the exact outage probability/transmitter density, rather than to nearest neighbor boun

Randomized Resource Allocaion in Decentralized Wireless Networks

Ad hoc networks and bluetooth systems operating over the unlicensed ISM band are in-stances of decentralized wireless networks. By definition, a decentralized network is com-posed of separate transmitter-receiver pairs where there is no central controller to assign the resources to the users. As such, resource allocation must be performed locally at each node. Users are anonymous to each other, i.e., they are not aware of each other's code-books. This implies that multiuser detection is not possible and users treat each other as noise. Multiuser interference is known to be the main factor that limits the achievable rates in such networks particularly in the high Signal-to-Noise Ratio (SNR) regime. Therefore, all users must follow a distributed signaling scheme such that the destructive effect of interference on each user is minimized, while the resources are fairly shared. In chapter 2 we consider a decentralized wireless communication network with a fixed number of frequency sub-bands to be shared among several transmitter-receiver pairs. It is assumed that the number of active users is a realization of a random variable with a given probability mass function. Moreover, users are unaware of each other's codebooks and hence, no multiuser detection is possible. We propose a randomized Frequency Hopping (FH) scheme in which each transmitter randomly hops over a subset of sub-bands from transmission slot to transmission slot. Assuming all users transmit Gaussian signals, the distribution of the noise plus interference is mixed Gaussian, which makes calculation of the mutual information between the transmitted and received signals of each user intractable. We derive lower and upper bounds on the mutual information of each user and demonstrate that, for large SNR values, the two bounds coincide. This observation enables us to compute the sum multiplexing gain of the system and obtain the optimum hopping strategy for maximizing this quantity. We compare the performance of the FH system with that of the Frequency Division (FD) system in terms of the following performance measures: average sum multiplexing gain and average minimum multiplexing gain per user. We show that (depending on the probability mass function of the number of active users) the FH system can offer a significant improvement in terms of the aforementioned measures. In the sequel, we consider a scenario where the transmitters are unaware of the number of active users in the network as well as the channel gains. Developing a new upper bound on the differential entropy of a mixed Gaussian random vector and using entropy power inequality, we obtain lower bounds on the maximum transmission rate per user to ensure a specified outage probability at a given SNR level. We demonstrate that the so-called outage capacity can be considerably higher in the FH scheme than in the FD scenario for reasonable distributions on the number of active users. This guarantees a higher spectral efficiency in FH compared to FD. Chapter 3 addresses spectral efficiency in decentralized wireless networks of separate transmitter-receiver pairs by generalizing the ideas developed in chapter 2. Motivated by random spreading in Code Division Multiple Access (CDMA), a signaling scheme is introduced where each user's code-book consists of two groups of codewords, referred to as signal codewords and signature codewords. Each signal codeword is a sequence of independent Gaussian random variables and each signature codeword is a sequence of independent random vectors constructed over a globally known alphabet. Using a conditional entropy power inequality and a key upper bound on the differential entropy of a mixed Gaussian random vector, we develop an inner bound on the capacity region of the decentralized network. To guarantee consistency and fairness, each user designs its signature codewords based on maximizing the average (with respect to a globally known distribution on the channel gains) of the achievable rate per user. It is demonstrated how the Sum Multiplexing Gain (SMG) in the network (regardless of the number of users) can be made arbitrarily close to the SMG of a centralized network with an orthogonal scheme such as Time Division (TD). An interesting observation is that in general the elements of the vectors in a signature codeword must not be equiprobable over the underlying alphabet in contrast to the use of binary Pseudo-random Noise (PN) signatures in randomly spread CDMA where the chip elements are +1 or -1 with equal probability. The main reason for this phenomenon is the interplay between two factors appearing in the expression of the achievable rate, i.e., multiplexing gain and the so-called interference entropy factor. In the sequel, invoking an information theoretic extremal inequality, we present an optimality result by showing that in randomized frequency hopping which is the main idea in the prevailing bluetooth devices in decentralized networks, transmission of independent signals in consecutive transmission slots is in general suboptimal regardless of the distribution of the signals. Finally, chapter 4 addresses a decentralized Gaussian interference channel consisting of two block-asynchronous transmitter-receiver pairs. We consider a scenario where the rate of data arrival at the encoders is considerably low and codewords of each user are transmitted at random instants depending on the availability of enough data for transmission. This makes the transmitted signals by each user look like scattered bursts along the time axis. Users are block-asynchronous meaning there exists a delay between their transmitted signal bursts. The proposed model for asynchrony assumes the starting point of an interference burst is uniformly distributed along the transmitted codeword of any user. There is also the possibility that each user does not experience interference on a transmitted codeword at all. Due to the randomness of delay, the channels are non-ergodic in the sense that the transmitters are unaware of the location of interference bursts along their transmitted codewords. In the proposed scheme, upon availability of enough data in its queue, each user follows a locally Randomized Masking (RM) strategy where the transmitter quits transmitting the Gaussian symbols in its codeword independently from symbol interval to symbol interval. An upper bound on the probability of outage per user is developed using entropy power inequality and a key upper bound on the differential entropy of a mixed Gaussian random variable. It is shown that by adopting the RM scheme, the probability of outage is considerably less than the case where both users transmit the Gaussian symbols in their codewords in consecutive symbol intervals, referred to as Continuous Transmission (CT)