Weighted Norms of Ambiguity Functions and Wigner Distributions

In this article new bounds on weighted p-norms of ambiguity functions and Wigner functions are derived. Such norms occur frequently in several areas of physics and engineering. In pulse optimization for Weyl--Heisenberg signaling in wide-sense stationary uncorrelated scattering channels for example it is a key step to find the optimal waveforms for a given scattering statistics which is a problem also well known in radar and sonar waveform optimizations. The same situation arises in quantum information processing and optical communication when optimizing pure quantum states for communicating in bosonic quantum channels, i.e. find optimal channel input states maximizing the pure state channel fidelity. Due to the non-convex nature of this problem the optimum and the maximizers itself are in general difficult find, numerically and analytically. Therefore upper bounds on the achievable performance are important which will be provided by this contribution. Based on a result due to E. Lieb, the main theorem states a new upper bound which is independent of the waveforms and becomes tight only for Gaussian weights and waveforms. A discussion of this particular important case, which tighten recent results on Gaussian quantum fidelity and coherent states, will be given. Another bound is presented for the case where scattering is determined only by some arbitrary region in phase space.Comment: 5 twocolumn pages,2 figures, accepted for 2006 IEEE International Symposium on Information Theory, typos corrected, some additional cites, legend in Fig.2 correcte

Pulse Shaping, Localization and the Approximate Eigenstructure of LTV Channels

In this article we show the relation between the theory of pulse shaping for WSSUS channels and the notion of approximate eigenstructure for time-varying channels. We consider pulse shaping for a general signaling scheme, called Weyl-Heisenberg signaling, which includes OFDM with cyclic prefix and OFDM/OQAM. The pulse design problem in the view of optimal WSSUS--averaged SINR is an interplay between localization and "orthogonality". The localization problem itself can be expressed in terms of eigenvalues of localization operators and is intimately connected to the concept of approximate eigenstructure of LTV channel operators. In fact, on the L_2-level both are equivalent as we will show. The concept of "orthogonality" in turn can be related to notion of tight frames. The right balance between these two sides is still an open problem. However, several statements on achievable values of certain localization measures and fundamental limits on SINR can already be made as will be shown in the paper.Comment: 6 pages, 2 figures, invited pape

On the Szeg\"o-Asymptotics for Doubly-Dispersive Gaussian Channels

We consider the time-continuous doubly-dispersive channel with additive Gaussian noise and establish a capacity formula for the case where the channel correlation operator is represented by a symbol which is periodic in time and fulfills some further integrability and smoothness conditions. The key to this result is a new Szeg\"o formula for certain pseudo-differential operators. The formula justifies the water-filling principle along time and frequency in terms of the time--continuous time-varying transfer function (the symbol).Comment: 5 pages, to be presented at ISIT 2011, minor typos corrected, references update

A Group-Theoretic Approach to the WSSUS Pulse Design Problem

We consider the pulse design problem in multicarrier transmission where the pulse shapes are adapted to the second order statistics of the WSSUS channel. Even though the problem has been addressed by many authors analytical insights are rather limited. First we show that the problem is equivalent to the pure state channel fidelity in quantum information theory. Next we present a new approach where the original optimization functional is related to an eigenvalue problem for a pseudo differential operator by utilizing unitary representations of the Weyl--Heisenberg group.A local approximation of the operator for underspread channels is derived which implicitly covers the concepts of pulse scaling and optimal phase space displacement. The problem is reformulated as a differential equation and the optimal pulses occur as eigenstates of the harmonic oscillator Hamiltonian. Furthermore this operator--algebraic approach is extended to provide exact solutions for different classes of scattering environments.Comment: 5 pages, final version for 2005 IEEE International Symposium on Information Theory; added references for section 2; corrected some typos; added more detailed discussion on the relations to quantum information theory; added some more references; added additional calculations as an appendix; corrected typo in III.

Stable Recovery from the Magnitude of Symmetrized Fourier Measurements

In this note we show that stable recovery of complex-valued signals $x\in\mathbb{C}^n$ up to global sign can be achieved from the magnitudes of $4n-1$ Fourier measurements when a certain "symmetrization and zero-padding" is performed before measurement ($4n-3$ is possible in certain cases). For real signals, symmetrization itself is linear and therefore our result is in this case a statement on uniform phase retrieval. Since complex conjugation is involved, such measurement procedure is not complex-linear but recovery is still possible from magnitudes of linear measurements on, for example, $(\Re(x),\Im(x))$.Comment: 4 pages, will be submitted to ICASSP1