H. O. Pollak

Bio: H. O. Pollak is an academic researcher from Columbia University. The author has contributed to research in topics: Connected Mathematics & Reform mathematics. The author has an hindex of 14, co-authored 36 publications receiving 4540 citations. Previous affiliations of H. O. Pollak include Telcordia Technologies & Bell Labs.

Prolate spheroidal wave functions, fourier analysis and uncertainty — II

Abstract: The theory developed in the preceding paper1 is applied to a number of questions about timelimited and bandlimited signals. In particular, if a finite-energy signal is given, the possible proportions of its energy in a finite time interval and a finite frequency band are found, as well as the signals which do the best job of simultaneous time and frequency concentration.

Steiner Minimal Trees

Abstract: A Steiner minimal tree for given points $A_1 , \cdots ,A_n $ in the plane is a tree which interconnects these points using lines of shortest possible total length. In order to achieve minimum lengt...

On the addressing problem for loop switching

Abstract: The methods used to perform the switching functions of the Bell System have been developed under the fundamental assumption that the holding time of the completed call is long compared to the time needed to set up the call. In considering certain forms of communication with and among computers the possibility arises that a message, with its destination at its head might thread its way through a communication network without awaiting the physical realization of a complete dedicated path before beginning on its journey. One such scheme has been proposed by J. R. Pierce and may be called “loop switching.” We imagine subscribers, perhaps best thought of as computer terminals or other data generating devices, on one-way loops. These “local” loops are connected by various switching points to one another as well as to other “regional” loops which are in turn connected to one another as well as to a “national” loop. If a message from one loop is destined for a subscriber on another loop it proceeds around the originating loop to a suitable switching point where it may choose to enter a different loop, this process continuing until the message reaches its destination. The question naturally comes up, how the message is to know which sequence of loops to follow. It would be desirable for the equipment at each junction to be able to apply a simple test to the destination addressing the loops which has several attractive features: (i) It permits an extremely simple routing strategy to be used by the messages in reaching their destmations. (ii) By using this strategy, a message will always take the shortest possible path between any two local loops in the same region. {iii) The method of addressing applies to any collection of loops, no matter hoio complex their interconnections. The addressing scheme we propose will be applied primarily to local loops where the mutual interconnections may be quite varied. If a certain amount of hierarchical structure is introduced into the regional and national loop structure, as suggested by J. R. Pierce1 it is possible to achieve addressings which are both compact and quite efficient.

On embedding graphs in squashed cubes

Abstract: We shall refer to d((s 1 ..... Sn) , (s~ ..... s~)) as the distance be! I tween the two n-tuples (s I ..... Sn) and (s 1 ..... s n) although, strictly speaking• this is an abuse of terminology since d does not satisfy the triangle inequality. For a connected graph G, the distance between two vertices v and v' in G, denoted by dG(V,V'), is defined to be the minimum number of edges in any path between v and v'. The following problem arose recently in connection with a data transmission scheme of J. R. Pierce [4].

Amplitude distribution of shot noise

Abstract: A shot noise, I(t), is a superposition of impulses occurring at random Poisson distributed times …, t −1 , t 0 , t 1 , t 2 , …. In the simplest case, if the impulses all have the same shape F(t), then $I(t)=\sum{\limits_{i}}F(t-t_i)$ . We study, in this and more general cases, the distribution function $Q(I)=Pr\[I(t)\ \leqq\ I\]$ . One of our results is an integral equation for Q(I). This yields explicit expressions for Q(I) in a number of cases, including F(t) = e−t; it also permits a computational technique which is applied to F(t) = e−t sin ωt for >> 1.

Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information

Abstract: This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f/spl isin/C/sup N/ and a randomly chosen set of frequencies /spl Omega/. Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set /spl Omega/? A typical result of this paper is as follows. Suppose that f is a superposition of |T| spikes f(t)=/spl sigma//sub /spl tau//spl isin/T/f(/spl tau/)/spl delta/(t-/spl tau/) obeying |T|/spl les/C/sub M//spl middot/(log N)/sup -1/ /spl middot/ |/spl Omega/| for some constant C/sub M/>0. We do not know the locations of the spikes nor their amplitudes. Then with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the /spl lscr//sub 1/ minimization problem. In short, exact recovery may be obtained by solving a convex optimization problem. We give numerical values for C/sub M/ which depend on the desired probability of success. Our result may be interpreted as a novel kind of nonlinear sampling theorem. In effect, it says that any signal made out of |T| spikes may be recovered by convex programming from almost every set of frequencies of size O(|T|/spl middot/logN). Moreover, this is nearly optimal in the sense that any method succeeding with probability 1-O(N/sup -M/) would in general require a number of frequency samples at least proportional to |T|/spl middot/logN. The methodology extends to a variety of other situations and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one- or two-dimensional) object from incomplete frequency samples - provided that the number of jumps (discontinuities) obeys the condition above - by minimizing other convex functionals such as the total variation of f.

Cognitive radio: brain-empowered wireless communications

Abstract: Cognitive radio is viewed as a novel approach for improving the utilization of a precious natural resource: the radio electromagnetic spectrum. The cognitive radio, built on a software-defined radio, is defined as an intelligent wireless communication system that is aware of its environment and uses the methodology of understanding-by-building to learn from the environment and adapt to statistical variations in the input stimuli, with two primary objectives in mind: /spl middot/ highly reliable communication whenever and wherever needed; /spl middot/ efficient utilization of the radio spectrum. Following the discussion of interference temperature as a new metric for the quantification and management of interference, the paper addresses three fundamental cognitive tasks. 1) Radio-scene analysis. 2) Channel-state estimation and predictive modeling. 3) Transmit-power control and dynamic spectrum management. This work also discusses the emergent behavior of cognitive radio.

On the use of windows for harmonic analysis with the discrete Fourier transform

Abstract: This paper makes available a concise review of data windows and their affect on the detection of harmonic signals in the presence of broad-band noise, and in the presence of nearby strong harmonic interference. We also call attention to a number of common errors in the application of windows when used with the fast Fourier transform. This paper includes a comprehensive catalog of data windows along with their significant performance parameters from which the different windows can be compared. Finally, an example demonstrates the use and value of windows to resolve closely spaced harmonic signals characterized by large differences in amplitude.

The wavelet transform, time-frequency localization and signal analysis

Abstract: Two different procedures for effecting a frequency analysis of a time-dependent signal locally in time are studied. The first procedure is the short-time or windowed Fourier transform; the second is the wavelet transform, in which high-frequency components are studied with sharper time resolution than low-frequency components. The similarities and the differences between these two methods are discussed. For both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time-frequency density. Finally, the notion of time-frequency localization is made precise, within this framework, by two localization theorems. >

Spectrum estimation and harmonic analysis

Abstract: In the choice of an estimator for the spectrum of a stationary time series from a finite sample of the process, the problems of bias control and consistency, or "smoothing," are dominant. In this paper we present a new method based on a "local" eigenexpansion to estimate the spectrum in terms of the solution of an integral equation. Computationally this method is equivalent to using the weishted average of a series of direct-spectrum estimates based on orthogonal data windows (discrete prolate spheroidal sequences) to treat both the bias and smoothing problems. Some of the attractive features of this estimate are: there are no arbitrary windows; it is a small sample theory; it is consistent; it provides an analysis-of-variance test for line components; and it has high resolution. We also show relations of this estimate to maximum-likelihood estimates, show that the estimation capacity of the estimate is high, and show applications to coherence and polyspectrum estimates.