Gianfranco Cariolaro

Multiplicity of fractional Fourier transforms and their relationships

The multiplicity of the fractional Fourier transform (FRT), which is intrinsic in any fractional operator, has been claimed by several authors, but never systematically developed. The paper starts with a general FRT definition, based on eigenfunctions and eigenvalues of the ordinary Fourier transform, which allows us to generate all possible definitions. The multiplicity is due to different choices of both the eigenfunction and the eigenvalue classes. A main result, obtained by a generalized form of the sampling theorem, gives explicit relationships between the different FRTs.

A unified framework for the fractional Fourier transform

The paper investigates the possibility for giving a general definition of the fractional Fourier transform (FRT) for all signal classes [one-dimensional (1-D) and multidimensional, continuous and discrete, periodic and aperiodic]. Since the definition is based on the eigenfunctions of the ordinary Fourier transform (FT), the preliminary conditions is that the signal domain/periodicity be the same as the FT domain/periodicity. Within these classes, a general FRT definition is formulated, and the FRT properties are established. In addition, the multiplicity (which is intrinsic in a fractional operator) is clearly developed. The general definition is checked in the case in which the FRT is presently available and, moreover, to establish the FRT in new classes of signals.

The fractional discrete cosine transform

The extension of the Fourier transform operator to a fractional power has received much attention in signal theory and is finding attractive applications. The paper introduces and develops the fractional discrete cosine transform (DCT) on the same lines, discussing multiplicity and computational aspects. Similarities and differences with respect to the fractional Fourier transform are pointed out.

Invited paper Analysis of codes and spectra calculations

Abstract The paper deals with the statistical analysis of the several kinds of signals encountered in digital transmission systems in which the data are coded by a line coder before being transmitted. In the preliminary sections, models of signals and systems are formulated. In particular, a general model is presented in which a line coder is split into three parts : a serial-to-parallel conversion, a finite-state sequential machine, and a parallel-to-serial conversion, where the fundamental coding function is described by a finite-state machine operating on blocks of source-symbols to produce blocks of code-symbols. On the basis of this model, the complete statistics of the coded sequence are evaluated in terms of the source probabilities. In the final part, consideration is given to spectral analysis, where both the continuous and the discrete part (spectral lines) of the spectral densities are calculated in closed form. The results obtained have a general validity and can be used for any line coding sc...

Method for defining a class of fractional operations

The fractional Fourier transform (FRT) permits a variety of associated fractional operations. This correspondence proposes a systematic method, based on the structure of the FRT, which not only provides unambiguous extensions of ordinary operations but permits writing the applicable expressions simply by inspection. The approach also exposes the possible paths for implementing such operations.

Error probability in digital fiber optic communication systems

In a digital fiber optic transmission system, during photodetection process, a shot noise is produced that is neither stationary nor independent of the digital message. The evaluation of the average error probability in the presence of such a message-dependent shot noise, of additive Gaussian noise, and of intersymbol interference is considered. Two methods of calculation are outlined: an exhaustive method and a Gram-Charlier series expansion method. The latter is preferred when the number of interferers is moderately large. Some numerical examples for binary independent-symbol transmission are presented.

Performance of quantum data transmission systems in the presence of thermal noise

In the literature the performance of quantum data transmission systems is usually evaluated in the absence of thermal noise. A more realistic approach taking into account the thermal noise is intrinsically more difficult because it requires dealing with Glauber coherent states in an infinite-dimensional space. In particular, the exact evaluation of the optimal measurement operators is a very difficult task, and numerical approximation is unavoidable. The paper faces the problem by approximating the P-representation of the noisy quantum states with a large but finite numbers of terms and applying to them the square root measurement (SRM) approach. Comparisons with cases where the exact solution are known show that the SRM approach gives quite accurate results. As application, the performance of quadrature amplitude modulation (QAM) and phase shift keying (PSK) systems is considered. In spite of the fact that the SRM approach is not optimal and overestimates the error probability, also in these cases the quantum detection maintains its superiority with respect to the classical homodyne detection.

Moments of correlated digital signals for error probability evaluation

In digital communication systems, the error probability in the presence of intersymbol interference (II) and additive noise may be calculated to any desired degree of accuracy by well-known approximation methods which avoid the exponential computation growth (with the number of interferers) inherent in an exhaustive method, on the condition that a fast technique for computing II moments is available. Such a technique is indeed available at present, but it is strongly limited by the assumption that the data symbols are mutually independent. In this paper, this limitation is removed, and a fast procedure is given for computing H moments of correlated digital signals. The computations grow linearly with the number of interferers. The assumption made is that correlated symbols are produced by a general finite-state sequential machine. As illustrative examples, the fast procedure is applied to bipolar and Franaszek MS-43 codes.

Classical and Quantum Information Theory

The purpose of this chapter is to provide an overview of Quantum Information theory starting from Classical Information Theory, with the aim to: (1) define information mathematically and quantitatively, (2) represent the information in an efficient way (through data compression) for storage and transmission, and (3) ensure the protection of information (through encoding) in the presence of noise and other impairments. In Classical Information theory, the above goals are accomplished in accordance to the laws of Classical Physics. In Quantum Information theory, they are based on quantum mechanical principles and are intrinsically richer than in their Classical counterpart, because of intriguing resources, as entanglement; also, they are more interesting and challenging.

Exact spectral evaluation of the family of digital pulse interval modulated signals

This paper presents a closed-form spectral evaluation of the family of digital pulse interval modulation (DPIM) schemes, where the information is contained in the relative distance between successive pulses. The framework for spectral evaluation is that of stationary symbol sequences derived from variable-length word sequences, which considers the issue of stationarization, a preliminary task to spectral analysis. For all the considered modulation schemes, the spectrum turns out to be a rational function of exp(j2/spl pi/fT), where T is the slot duration. Moreover, spectral lines occur at integer multiples of 1/T, but occasionally may also occur at submultiples of 1/T.