Nikolaos M. Freris

Fundamental Limits on Synchronizing Clocks Over Networks

We characterize what is feasible concerning clock synchronization in wireline or wireless networks. We consider a net work of n nodes, equipped with affine clocks relative to a designated clock that exchange packets subject to link delays. Determining all unknown parameters, i.e., skews and offsets of all the clocks as well as the delays of all the communication links, is impossible. All nodal skews, as well as all round-trip delays between every pair of nodes, can be determined correctly. Also, every transmitting node can predict precisely the time indicated by the receivers clock at which it receives the packet. However, the vector of unknown link delays and clock offsets can only be determined up to an (n - 1)-dimensional subspace, with each degree of freedom corresponding to the offset of one of the (n - 1) clocks. Invoking causality, that packets cannot be received before they are transmitted, the uncertainty set can be reduced to a polyhedron. We also investigate structured models for link delays as the sum of a transmitter-dependent delay, a receiver-dependent delay, and a known propagation delay, and identify conditions which permit a unique solution, and conditions under which the number of the residual degrees of freedom is independent of the network size. For receiver-receiver synchronization, where only receipt times are available, but no time-stamping is done by the sender, all nodal skews can still be determined, but delay differences between neighboring communication links with a common sender can only be characterized up to an affine transformation of the (n - 1) un known offsets. Moreover, causality does not help reduce the uncertainty set.

Fundamentals of Large Sensor Networks: Connectivity, Capacity, Clocks, and Computation

Sensor networks potentially feature large numbers of nodes. The nodes can monitor and sense their environment over time, communicate with each other over a wireless network, and process information that they exchange with each other. They differ from data networks in that the network as a whole may be designed for a specific application. We study the theoretical foundations of such large-scale sensor networks. We address four fundamental organizational and operational issues related to large sensor networks: connectivity, capacity, clocks, and function computation. To begin with, a sensor network must be connected so that information can indeed be exchanged between nodes. The connectivity graph of an ad hoc network is modeled as a random graph and the critical range for asymptotic connectivity is determined, as well as the critical number of neighbors that a node needs to connect to. Next, given connectivity, we address the issue of how much data can be transported over the sensor network. We present fundamental bounds on capacity under several models, as well as architectural implications for how wireless communication should be organized. Temporal information is important both for the applications of sensor networks as well as their operation. We present fundamental bounds on the synchronizability of clocks in networks, and also present and analyze algorithms for clock synchronization. Finally, we turn to the issue of gathering relevant information, which sensor networks are designed to do. One needs to study optimal strategies for in-network aggregation of data, in order to reliably compute a composite function of sensor measurements, as well as the complexity of doing so. We address the issue of how such computation can be performed efficiently in a sensor network and the algorithms for doing so, for some classes of functions.

Randomized Extended Kaczmarz For Solving Least Squares

We present a randomized iterative algorithm that exponentially converges in the mean square to the minimum

Fundamental limits on synchronization of affine clocks in networks

\ell_2

A model-based approach to clock synchronization

-norm least squares solution of a given linear system of equations. The expected number of arithmetic operations required to obtain an estimate of given accuracy is proportional to the squared condition number of the system multiplied by the number of nonzero entries of the input matrix. The proposed algorithm is an extension of the randomized Kaczmarz method that was analyzed by Strohmer and Vershynin.

Fast distributed smoothing of relative measurements

We present some impossibility as well as feasibility results on clock synchronization in wireline or wireless networks. We consider a network of n nodes with affine clocks, where one node is taken as a reference, and each other nodes clock is described by a skew (relative speedup with respect to the reference clock), and an offset (say) at time 0 with respect to the reference clock. In order to establish impossibility results, we allow for noiseless communication of messages that may contain any information that the transmitting node knows about current or past packets that the node has sent or received. The synchronization problem consists of estimating all the unknown parameters, skews and offsets of all the clocks, as well as the delays of all the communication links. We show that, in general, estimation of all unknown parameters is impossible. We show that all nodal skews, as well as all round trip delays between every pair of nodes, can be estimated correctly. However, the unknown link delays and clock offsets can only be determined up to an (n - 1)-dimensional subspace. Each degree of freedom in this subspace corresponds exactly to the offset of one of the n-1 clocks with respect to the reference nodes clock. On the positive side, we show that every transmitting node can predict precisely the time indicated by the receivers clock at the instant it receives the packet. If we further invoke causality, i.e., that packets cannot be received before they are transmitted, the uncertainty set above can be reduced to a compact polyhedron in Rn-1, which we describe analytically. We also consider a specific model for delays as the sum of a transmitter-dependent delay, a receiver-dependent delay and a propagation delay where the latter is known. We identify conditions on the transmission and reception delays, for which estimation admits a unique solution.

Finite rate of innovation based modeling and compression of ECG signals

In a network of clocks, we consider a given reference node to determine the time evolution t. We introduce and analyze a stochastic model for clocks, in which the relative speedup of a clock, called the skew, is characterized by some given stochastic process. We study the problem of synchronizing clocks in a network, which amounts to estimating the instantaneous relative skews and relative offsets by exchange of time-stamped packets across the links of the network. We present a scheme for obtaining measurements in a communication link. We develop an algorithm for optimal filtering of measurements across a link (i; j) in order to estimate the logarithm of the relative speedup of node j with respect to node i, and we further study some implementation issues. We also present a scheme for pairwise offset estimation based on skew estimates. We study the properties of our algorithms and provide theoretical guarantees on their performance. We also develop an online centralized model-based asynchronous algorithm for optimal filtering of the time-stamps in the entire network, and an efficient distributed suboptimal scheme.

Resource Allocation for Multihomed Scalable Video Streaming to Multiple Clients

We consider the problem of estimation from noisy relative measurements in a network. In previous work, a distributed scheme for obtaining least-squares (LS) estimates was developed based on the Jacobi algorithm; in a synchronous version, the algorithm was shown to converge exponentially and bounds on the rate of convergence have been obtained. In this paper, we design and analyze a new class of distributed asynchronous smoothing algorithms based on a randomized version of Kaczmarz algorithm for solving linear systems. One of the proposed schemes applies Randomized Kaczmarz directly to the noisy linear system, whereas the other one operates on the normal equations for LS estimation. We analyze the expected convergence rate of the proposed algorithms depending solely on properties of the network topology. Inspired by the analytical insights, we propose a distributed smoothing algorithm, namely Randomized Kaczmarz Over-smoothing (RKO), which has demonstrated significant improvement over existing protocols in terms of both convergence speedup and energy savings.

Controlled random access MAC for network utility maximization in wireless networks

Mobile health is gaining increasing importance for society and the quest for new power efficient devices sampling biosignals is becoming critical. We discuss a new scheme called Variable Pulse Width Finite Rate of Innovation (VPW-FRI) to model and compress ECG signals. This technique generalizes classical FRI estimation to enable the use of a sum of asymmetric Cauchy-based pulses for modeling electrocardiogram (ECG) signals. We experimentally show that VPW-FRI indeed models ECG signals with increased accuracy compared to current standards. In addition, we study the compression efficiency of the method: compared with various widely used compression schemes, we showcase improvements in terms of compression efficiency as well as sampling rate.

Rate control and stream adaptation for scalable video streaming over multiple access networks

We consider multihomed scalable video streaming, where videos are transmitted by a single server to multiple clients over heterogeneous access networks. The specific problem that we address is to determine which video packets to transmit over each network, in order to minimize a cost function of the expected video distortion at the clients. We present a network model and a video model that capture the network conditions and video characteristics, respectively. We develop an integer program for deterministic packet scheduling. We propose different cost functions in order to provide service differentiation and address fairness among users. We propose several suboptimal convex problems for randomized packet scheduling, and study their performance and complexity. We propose an algorithm that yields a good performance and is suitable for real-time applications. We conduct extensive trace-driven simulations to evaluate the proposed algorithms using real network conditions and scalable video streams. The simulation results show that the proposed algorithm: (i) outperforms the rate control algorithms defined in the Datagram Congestion Control Protocol (DCCP) by about 10 dB, (ii) results in video quality, of 4.33 dB and 1.84 dB higher than the two heuristics developed in [1], (iii) runs efficiently, up to six times faster than one of the heuristics, and (iv) indeed can provide service differentiation among users.