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Edwin Hammerich

Publications and source records attributed to Edwin Hammerich.

7 recordsLinked to original sources
arXiv2017

Capacity and Normalized Optimal Detection Error in Gaussian Channels

For vector Gaussian channels, a precise differential connection between channel capacity and a quantity termed normalized optimal detection error (NODE) is presented. Then, this C-NODE relationship is extended to continuous-time Gaussian channels drawing on a waterfilling characterization recently found for the capacity of continuous-time linear time-varying channels. In the latter case, the C-NODE relationship becomes asymptotic in nature. In either case, the C-NODE relationship is compared with the I-MMSE relationship due to Guo et al. connecting mutual information in Gaussian channels with the minimum mean-square error (MMSE) of estimation theory.

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arXiv2015

Waterfilling Theorems for Linear Time-Varying Channels and Related Nonstationary Sources

The capacity of the linear time-varying (LTV) channel, a continuous-time LTV filter with additive white Gaussian noise, is characterized by waterfilling in the time-frequency plane. Similarly, the rate distortion function for a related nonstationary source is characterized by reverse waterfilling in the time-frequency plane. Constraints on the average energy or on the squared-error distortion, respectively, are used. The source is formed by the white Gaussian noise response of the same LTV filter as before. The proofs of both waterfilling theorems rely on a Szego theorem for a class of operators associated with the filter. A self-contained proof of the Szego theorem is given. The waterfilling theorems compare well with the classical results of Gallager and Berger. In the case of a nonstationary source, it is observed that the part of the classical power spectral density is taken by the Wigner-Ville spectrum. The present approach is based on the spread Weyl symbol of the LTV filter, and is asymptotic in nature. For the spreading factor, a lower bound is suggested by means of an uncertainty inequality.

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arXiv2014

Waterfilling Theorems in the Time-Frequency Plane for the Heat Channel and a Related Source

The capacity of the heat channel, a linear time-varying (LTV) filter with additive white Gaussian noise (AWGN), is characterized by waterfilling in the time-frequency plane. Similarly, the rate distortion function for a related nonstationary source is characterized by reverse waterfilling in the time-frequency plane. The source is formed by the white Gaussian noise response of the same LTV filter as before. The proofs of both waterfilling theorems rely on a specific Szego theorem for a positive definite operator associated with the filter. An essentially self-contained proof of the Szego theorem is given. The waterfilling theorems compare well with classical results of Gallager and Berger. In case of the nonstationary source it is observed that the part of the classical power spectral density (PSD) is taken by the Wigner-Ville spectrum (WVS).

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arXiv2014

On the Capacity of the Heat Channel, Waterfilling in the Time-Frequency Plane, and a C-NODE Relationship

The heat channel is defined by a linear time-varying (LTV) filter with additive white Gaussian noise (AWGN) at the filter output. The continuous-time LTV filter is related to the heat kernel of the quantum mechanical harmonic oscillator, so the name of the channel. The channel's capacity is given in closed form by means of the Lambert W function. Also a waterfilling theorem in the time-frequency plane for the capacity is derived. It relies on a specific Szego theorem for which an essentially self-contained proof is provided. Similarly, the rate distortion function for a related nonstationary source is given in closed form and a (reverse) waterfilling theorem in the time-frequency plane is derived. Finally, a second closed-form expression for the capacity of the heat channel based on the detected perturbed filter output signals is presented. In this context, a precise differential connection between channel capacity and the normalized optimal detection error (NODE) is revealed. This C-NODE relationship is compared with the well-known I-MMSE relationship connecting mutual information with the minimum mean-square error (MMSE) of estimation theory.

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arXiv2009

Design of Pulse Shapes Based on Sampling with Gaussian Prefilter

Two new pulse shapes for communications are presented. The first pulse shape generates a set of pulses without intersymbol interference (ISI) or ISI-free for short. In the neighborhood of the origin it is similar in shape to the classical cardinal sine function but is of exponential decay at infinity. This pulse shape is identical to the interpolating function of a generalized sampling theorem with Gaussian prefilter. The second pulse shape is obtained from the first pulse shape by spectral factorization. Besides being also of exponential decay at infinity, it has a causal appearance since it is of superexponential decay for negative times. It is closely related to the orthonormal generating function considered earlier by Unser in the context of shift-invariant spaces. This pulse shape is not ISI-free but it generates a set of orthonormal pulses. The second pulse shape may also be used to define a receive matched filter so that at the filter output the ISI-free pulses of the first kind are recovered.

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arXiv2009

A Generalized Sampling Theorem for Frequency Localized Signals

A generalized sampling theorem for frequency localized signals is presented. The generalization in the proposed model of sampling is twofold: (1) It applies to various prefilters effecting a "soft" bandlimitation, (2) an approximate reconstruction from sample values rather than a perfect one is obtained (though the former might be "practically perfect" in many cases). For an arbitrary finite-energy signal the frequency localization is performed by a prefilter realizing a crosscorrelation with a function of prescribed properties. The range of the filter, the so-called localization space, is described in some detail. Regular sampling is applied and a reconstruction formula is given. For the reconstruction error a general error estimate is derived and connections between a critical sampling interval and notions of "soft bandwidth" for the prefilter are indicated. Examples based on the sinc-function, Gaussian functions and B-splines are discussed.

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