Brendan Farrell

Strong divergence of reconstruction procedures for the Paley–Wiener space PW(1_π) and the Hardy space H^1

Previous results on certain sampling series have left open if divergence only occurs for certain subsequences or, in fact, in the limit. Here we prove that divergence occurs in the limit. We consider three canonical reconstruction methods for functions in the Paley–Wiener space PW^1_π. For each of these we prove an instance when the reconstruction diverges in the limit. This is a much stronger statement than previous results that provide only lim sup divergence. We also address reconstruction for functions in the Hardy space H^1 and show that for any subsequence of the natural numbers there exists a function in H^1 for which reconstruction diverges in lim sup. For two of these sampling series we show that when divergence occurs, the sampling series has strong oscillations so that the maximum and the minimum tend to positive and negative infinity. Our results are of interest in functional analysis because they go beyond the type of result that can be obtained using the Banach–Steinhaus Theorem. We discuss practical implications of this work; in particular the work shows that methods using specially chosen subsequences of reconstructions cannot yield convergence for the Paley–Wiener Space PW^1_π.

Local Eigenvalue Density for General MANOVA Matrices

AbstractWe consider random n×n matrices of the form

PAPR and the Density of Information Bearing Signals in OFDM

\begin{aligned} \left( XX^*+YY^*\right)^{-\frac{1}{2}}YY^*\left( XX^*+YY^*\right )^{-\frac{1}{2}} , \end{aligned}

Expected Supremum of a Random Linear Combination of Shifted Kernels

where X and Y have independent entries with zero mean and variance one. These matrices are the natural generalization of the Gaussian case, which are known as MANOVA matrices and which have joint eigenvalue density given by the third classical ensemble, the Jacobi ensemble. We show that, away from the spectral edge, the eigenvalue density converges to the limiting density of the Jacobi ensemble even on the shortest possible scales of order 1/n (up to logn factors). This result is the analogue of the local Wigner semicircle law and the local Marchenko-Pastur law for general MANOVA matrices.

PAPR for OFDM and the proportion of information bearing signals for tone reservation

High peak values of transmission signals in wireless communication systems lead to wasteful energy consumption and out-of-band radiation. However, reducing peak values generally comes at the cost of some other resource. We provide a theoretical contribution toward understanding the relationship between peak value reduction and the resulting cost in information rates. In particular, we address the relationship between peak values and the proportion of transmission signals allocated for information transmission when one is using a strategy known as tone reservation. We show that when tone reservation is used in both OFDM and DS-CDMA systems, if a peak-to-average power ratio criterion is always satisfied, then the proportion of transmission signals that may be allocated for information transmission must tend to zero. We investigate properties of these two systems for sets of both finite and infinite cardinalities. We present properties that OFDM and DS-CDMA share in common as well as ways in which they fundamentally differ.

Local spectrum of truncations of Kronecker products of Haar distributed unitary matrices

We address the peak-to-average power ratio (PAPR) of transmission signals in OFDM and consider the performance of tone reservation for reduction of the PAPR. Tone reservation is unique among methods for reducing PAPR, because it does not affect information bearing coefficients and involves no additional coordination of transmitter and receiver. It is shown that if the OFDM system always satisfies a given peak-to-average power ratio constraint, then the efficiency of the system, defined as the ratio of the number of tones used for information to the entire number of tones used, must converge to zero as the total number of tones increases. More generally, we investigate and provide insight into a tradeoff between optimal signal and information properties for OFDM systems and show that it is necessary to use very small subsets of the available signals to achieve PAPR reduction using tone reservation.

Peak Behavior and Information Rates for Orthonormal Systems

We consider linear time-varying channels with additive white Gaussian noise. For a large class of such channels we derive rigorous estimates of the eigenvalues of the correlation matrix of the effective channel in terms of the sampled time-varying transfer function and, thus, provide a theoretical justification for a relationship that has been frequently observed in the literature. We then use this eigenvalue estimate to derive an estimate of the mutual information of the channel. Our approach is constructive and is based on a careful balance of the tradeoff between approximate operator diagonalization, signal dimension loss, and accuracy of eigenvalue estimates.

Limiting Empirical Singular Value Distribution of Restrictions of Discrete Fourier Transform Matrices

We address the expected supremum of a linear combination of shifts of the sinc kernel with random coefficients. When the coefficients are Gaussian, the expected supremum is of order