Tracking Stopping Times Through Noisy Observations

A novel quickest detection setting is proposed which is a generalization of the well-known Bayesian change-point detection model. Suppose \{(X_i,Y_i)\}_{i\geq 1} is a sequence of pairs of random variables, and that S is a stopping time with respect to \{X_i\}_{i\geq 1}. The problem is to find a stopping time T with respect to \{Y_i\}_{i\geq 1} that optimally tracks S, in the sense that T minimizes the expected reaction delay E(T-S)^+, while keeping the false-alarm probability P(T<S) below a given threshold \alpha \in [0,1]. This problem formulation applies in several areas, such as in communication, detection, forecasting, and quality control. Our results relate to the situation where the X_i's and Y_i's take values in finite alphabets and where S is bounded by some positive integer \kappa. By using elementary methods based on the analysis of the tree structure of stopping times, we exhibit an algorithm that computes the optimal average reaction delays for all \alpha \in [0,1], and constructs the associated optimal stopping times T. Under certain conditions on \{(X_i,Y_i)\}_{i\geq 1} and S, the algorithm running time is polynomial in \kappa.Comment: 19 pages, 4 figures, to appear in IEEE Transactions on Information Theor

Bits through queues with feedback

In their $1996$ paper Anantharam and Verd\'u showed that feedback does not increase the capacity of a queue when the service time is exponentially distributed. Whether this conclusion holds for general service times has remained an open question which this paper addresses. Two main results are established for both the discrete-time and the continuous-time models. First, a sufficient condition on the service distribution for feedback to increase capacity under FIFO service policy. Underlying this condition is a notion of weak feedback wherein instead of the queue departure times the transmitter is informed about the instants when packets start to be served. Second, a condition in terms of output entropy rate under which feedback does not increase capacity. This condition is general in that it depends on the output entropy rate of the queue but explicitly depends neither on the queue policy nor on the service time distribution. This condition is satisfied, for instance, by queues with LCFS service policies and bounded service times

Optimal Sequential Frame Synchronization

We consider the `one-shot frame synchronization problem' where a decoder wants to locate a sync pattern at the output of a channel on the basis of sequential observations. We assume that the sync pattern of length N starts being emitted at a random time within some interval of size A, that characterizes the asynchronism level between the transmitter and the receiver. We show that a sequential decoder can optimally locate the sync pattern, i.e., exactly, without delay, and with probability approaching one as N tends to infinity, if and only if the asynchronism level grows as O(exp(N*k)), with k below the `synchronization threshold,' a constant that admits a simple expression depending on the channel. This constant is the same as the one that characterizes the limit for reliable asynchronous communication, as was recently reported by the authors. If k exceeds the synchronization threshold, any decoder, sequential or non-sequential, locates the sync pattern with an error that tends to one as N tends to infinity. Hence, a sequential decoder can locate a sync pattern as well as the (non-sequential) maximum likelihood decoder that operates on the basis of output sequences of maximum length A+N-1, but with much fewer observations.Comment: 6 page

Energy and Sampling Constrained Asynchronous Communication

The minimum energy, and, more generally, the minimum cost, to transmit one bit of information has been recently derived for bursty communication when information is available infrequently at random times at the transmitter. This result assumes that the receiver is always in the listening mode and samples all channel outputs until it makes a decision. If the receiver is constrained to sample only a fraction f>0 of the channel outputs, what is the cost penalty due to sparse output sampling? Remarkably, there is no penalty: regardless of f>0 the asynchronous capacity per unit cost is the same as under full sampling, ie, when f=1. There is not even a penalty in terms of decoding delay---the elapsed time between when information is available until when it is decoded. This latter result relies on the possibility to sample adaptively; the next sample can be chosen as a function of past samples. Under non-adaptive sampling, it is possible to achieve the full sampling asynchronous capacity per unit cost, but the decoding delay gets multiplied by 1/f. Therefore adaptive sampling strategies are of particular interest in the very sparse sampling regime.Comment: Submitted to the IEEE Transactions on Information Theor