Geometrical interpretations of Bäcklund transformations and certain types of partial differential equations : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University

Page 37 is missing from the original copy.Gauss' Theorema Egregium contains a partial differential equation relating the Gaussian curvature K to components of the metric tensor and its derivatives. Well known partial differential equations such as the Schrödinger equation and the sine-Gordon equation correspond to this PDE for special choices of K and special coördinate systems. The sine-Gordon equation, for example, can be derived via Gauss' equation for K = –1 using the Tchebychef net as a coördinate system. In this thesis we consider a special class of Bäcklund Transformations which correspond to coördinate transformations on surfaces having a specified Gaussian curvature. These transformations lead to Gauss' PDE in different forms and provide a method for solving certain classes of non-linear second order partial differential equations. In addition, we develop a more systematic way to obtain a coordinate system for a more general class of PDE, such that this PDE corresponds to the Gauss equation

Handoff Rate and Coverage Analysis in Multi-tier Heterogeneous Networks

This paper analyzes the impact of user mobility in multi-tier heterogeneous networks. We begin by obtaining the handoff rate for a mobile user in an irregular cellular network with the access point locations modeled as a homogeneous Poisson point process. The received signal-to-interference-ratio (SIR) distribution along with a chosen SIR threshold is then used to obtain the probability of coverage. To capture potential connection failures due to mobility, we assume that a fraction of handoffs result in such failures. Considering a multi-tier network with orthogonal spectrum allocation among tiers and the maximum biased average received power as the tier association metric, we derive the probability of coverage for two cases: 1) the user is stationary (i.e., handoffs do not occur, or the system is not sensitive to handoffs); 2) the user is mobile, and the system is sensitive to handoffs. We derive the optimal bias factors to maximize the coverage. We show that when the user is mobile, and the network is sensitive to handoffs, both the optimum tier association and the probability of coverage depend on the user's speed; a speed-dependent bias factor can then adjust the tier association to effectively improve the coverage, and hence system performance, in a fully-loaded network.Comment: Accepted for publication in the IEEE Transactions on Wireless Communication

Grassmannian Beamforming for MIMO Amplify-and-Forward Relaying

In this paper, we derive the optimal transmitter/ receiver beamforming vectors and relay weighting matrix for the multiple-input multiple-output amplify-and-forward relay channel. The analysis is accomplished in two steps. In the first step, the direct link between the transmitter (Tx) and receiver (Rx) is ignored and we show that the transmitter and the relay should map their signals to the strongest right singular vectors of the Tx-relay and relay-Rx channels. Based on the distributions of these vectors for independent identically distributed (i.i.d.) Rayleigh channels, the Grassmannian codebooks are used for quantizing and sending back the channel information to the transmitter and the relay. The simulation results show that even a few number of bits can considerably increase the link reliability in terms of bit error rate. For the second step, the direct link is considered in the problem model and we derive the optimization problem that identifies the optimal Tx beamforming vector. For the i.i.d Rayleigh channels, we show that the solution to this problem is uniformly distributed on the unit sphere and we justify the appropriateness of the Grassmannian codebook (for determining the optimal beamforming vector), both analytically and by simulation. Finally, a modified quantizing scheme is presented which introduces a negligible degradation in the system performance but significantly reduces the required number of feedback bits.Comment: Submitted to IEEE Journal of Selected Areas in Communications, Special Issue on Exploiting Limited Feedback in Tomorrows Wireless Communication Network

Bounds on the Capacity of ASK Molecular Communication Channels with ISI

There are now several works on the use of the additive inverse Gaussian noise (AIGN) model for the random transit time in molecular communication~(MC) channels. The randomness invariably causes inter-symbol interference (ISI) in MC, an issue largely ignored or simplified. In this paper we derive an upper bound and two lower bounds for MC based on amplitude shift keying (ASK) in presence of ISI. The Blahut-Arimoto algorithm~(BAA) is modified to find the input distribution of transmitted symbols to maximize the lower bounds. Our results show that over wide parameter values the bounds are close.Comment: 7 pages, 4 figures, Accepted in IEEE GLOBECOM 201

Information Rates of ASK-Based Molecular Communication in Fluid Media

This paper studies the capacity of molecular communications in fluid media, where the information is encoded in the number of transmitted molecules in a time-slot (amplitude shift keying). The propagation of molecules is governed by random Brownian motion and the communication is in general subject to inter-symbol interference (ISI). We first consider the case where ISI is negligible and analyze the capacity and the capacity per unit cost of the resulting discrete memoryless molecular channel and the effect of possible practical constraints, such as limitations on peak and/or average number of transmitted molecules per transmission. In the case with a constrained peak molecular emission, we show that as the time-slot duration increases, the input distribution achieving the capacity per channel use transitions from binary inputs to a discrete uniform distribution. In this paper, we also analyze the impact of ISI. Crucially, we account for the correlation that ISI induces between channel output symbols. We derive an upper bound and two lower bounds on the capacity in this setting. Using the input distribution obtained by an extended Blahut-Arimoto algorithm, we maximize the lower bounds. Our results show that, over a wide range of parameter values, the bounds are close.Comment: 31 pages, 8 figures, Accepted for publication on IEEE Transactions on Molecular, Biological, and Multi-Scale Communication

Large-Scale MIMO versus Network MIMO for Multicell Interference Mitigation

This paper compares two important downlink multicell interference mitigation techniques, namely, large-scale (LS) multiple-input multiple-output (MIMO) and network MIMO. We consider a cooperative wireless cellular system operating in time-division duplex (TDD) mode, wherein each cooperating cluster includes $B$ base-stations (BSs), each equipped with multiple antennas and scheduling $K$ single-antenna users. In an LS-MIMO system, each BS employs $BM$ antennas not only to serve its scheduled users, but also to null out interference caused to the other users within the cooperating cluster using zero-forcing (ZF) beamforming. In a network MIMO system, each BS is equipped with only $M$ antennas, but interference cancellation is realized by data and channel state information exchange over the backhaul links and joint downlink transmission using ZF beamforming. Both systems are able to completely eliminate intra-cluster interference and to provide the same number of spatial degrees of freedom per user. Assuming the uplink-downlink channel reciprocity provided by TDD, both systems are subject to identical channel acquisition overhead during the uplink pilot transmission stage. Further, the available sum power at each cluster is fixed and assumed to be equally distributed across the downlink beams in both systems. Building upon the channel distribution functions and using tools from stochastic ordering, this paper shows, however, that from a performance point of view, users experience better quality of service, averaged over small-scale fading, under an LS-MIMO system than a network MIMO system. Numerical simulations for a multicell network reveal that this conclusion also holds true with regularized ZF beamforming scheme. Hence, given the likely lower cost of adding excess number of antennas at each BS, LS-MIMO could be the preferred route toward interference mitigation in cellular networks.Comment: 13 pages, 7 figures; IEEE Journal of Selected Topics in Signal Processing, Special Issue on Signal Processing for Large-Scale MIMO Communication