Quantum detection in an N-dimensional Hilbert space H involves quantum states and corresponding measure ment operators which span an r-dimensional subspace U of H, with r ≤ N. Quantum detection could be restricted to this subspace, but the operations in U are still redundant, since the kets have N components. By applying the singular-value decomposition to the state matrix, it is possible to perform a compression from the subspace U onto a "compressed" space U̅, where the redundancy is removed and kets are represented by r components. The detection can be perfectly reformulated in the "compressed" space, without loss of information, with a greatly reduced complexity. The compression is particularly attractive when r ≪ N, as shown with an example of application to quantum optical communications.

This paper deals with the spectral analysis of multi-h continuous phase modulation (CPM) signals. A new solution to the problem is obtained by identifying an inner hidden Markov model in the CPM signal, which can be accomplished by expressing the modulator as the cascade of a sequential machine (SM), and a bank of PAM modulators. An in-depth study of the resulting Markov chain shows that it is both cyclostationary and reducible, and proper tools are identified to address these peculiarities. As a result, the multi-h CPM spectrum is expressed as a combination of Fourier transforms of portions of a CPM signal, thus allowing an efficient numerical evaluation through fast Fourier transforms (FFTs).

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.

The paper deals with quantum pulse position modulation (PPM), both in the absence (pure states) and in the presence (mixed states) of thermal noise, using the Glauber representation of coherent laser radiation. The objective is to find optimal (or suboptimal) measurement operators and to evaluate the corresponding error probability. For PPM, the correct formulation of quantum states is given by the tensorial product of m identical Hilbert spaces, where m is the PPM order. The presence of mixed states, due to thermal noise, generates an optimization problem involving matrices of huge dimensions, which already for 4-PPM, are of the order of ten thousand. To overcome this computational complexity, the currently available methods of quantum detection, which are based on explicit results, convex linear programming and square root measurement, are compared to find the computationally less expensive one. In this paper a fundamental role is played by the geometrically uniform symmetry of the quantum PPM format. The evaluation of error probability confirms the vast superiority of the quantum detection over its classical counterpart.

A method for designing a finite automaton whose output exhibits a given rational power spectral density R(z) belonging to a particular class, is presented. If the automaton input is composed of independent and identically distributed symbols, its state process is a homogeneous Markov chain whose transition probability matrix pi may by obtained from the input probability mass function and the state transition function. Since the poles of R(z) only depend on pi whereas its zeros depend on the matrix A specifying the output function, a matrix pi with the desired eigenvalues (and perhaps additional ones) is first derived and, the matrix A is determined so as to ensure the realization of the desired zeros (as well as the cancellation of the additional poles possibly introduced in the first step). The method exploits the properties of circulant matrices; in particular, a sufficient condition is provided under which a circular matrix with given eigenvalues (ordered in a Hermitian sequence) is stochastic.>

A pulse stuffing technique (used for TDM) in which both positive and negative stuffings are allowed with two stuffing thresholds periodically modulated by a sawtooth waveform is discussed. A suitable mathematical model of the technique and a frequency domain analysis of the inherent pulse stuffing jitter are provided. It is found that the approach reduces the peak-to-peak pulse stuffing jitter according to the inverse of the threshold modulation period. The analytical results are confirmed by numerical simulations.>

Codes whose spectral density and its lower order derivatives vanish at zero frequency are investigated, and necessary and sufficient conditions for the spectral properties are given. In particular, for the most common balanced alphabets, a complete characterization is furnished for the class of symbol-by-symbol codes with vanishing spectrum and second derivative. The fundamental results are extended to block codes, and suitable criteria for their design are given. Specific examples of binary and ternary block codes are presented; their performance in terms of low power content at low frequencies compares advantageously with that of some conventional codes.>

A method is proposed for launching and extracting the timing wave in a PAM digital transmission system, which combines the simplicity of linear self-timing, the economy in bandwidth of nonlinear self-timing, and the high performance of the added timing wave approach. The method requires that the digital signal is cyclostationary with a period which is an even multiple of the symbol periodT; the simplest solution for deriving such a signal from a stationary message is to apply it to a periodically time-varying system, as a memoryless block coder. With reference to such a scheme we derive the spectrum of the timing jitter. Very simple conditions are found on block encoding in order that the encoded message exhibits the desired characteristics. Finally, ternary codes are discussed and the performance of a specific code is evaluated and compared to that of conventional extraction schemes.

The spectral analysis is performed for the stochastic sequence\{b_{n}\} = \{h(a_{n}, a_{n-1},\cdots, a_{n-M+1})\}, where\{a_{n}\}is an independent identically distributed digital sequence andhis an arbitrary complex function. The computation of the spectrum is based on a general matrix method for the spectral analysis of memoryless functions of Markov chains. The approach employs a recursive algorithm, whose computational time and storage requirements are sufficiently small for the algorithm to be implemented on a modern general purpose computer, even for moderately high values of both the size of the source alphabet and the parameterM. The method is also extended to the case in which the input sequence\{a_{n}\}is a stationary finite Markov chain.

In pulse-amplitude modulation (PAM) digital transmission systems line encoding is used for shaping the spectrum of the encoded symbol sequence to suit the frequency characteristics of the transmission channel. In particular, it is often required that the encoded symbol sequence have a zero mean and spectral density vanishing at zero frequency. We show that the finite running digital sum condition is a necessary and sufficient condition for this to occur. The result holds in particular for alphabetic codes, which are the most widely used line codes.

This paper deals with the spectral analysis of digital processes obtained through memoryless functions of stationary Markov chains. Under the assumption that the Markov chain is ergodic, not necessarily acyclic, closed form formulas are derived for both the discrete part (spectral lines) and the continuous part of the spectral density. The results are applied at the output of a finite state sequential machine driven by a stationary Markov chain, which represents a general model of several situations of engineering interest. Finally, the theory is used to evaluate the spectrum of encoded and modulated digital signals.

The paper deals with the spectral analysis of both phase and frequency modulated digital signals with correlated messages, where correlation is obtained by encoding. Also, a possible general filtering of the baseband digital signal is taken into account. The method of computation is fast, since the computational complexity increases linearly with the baseband pulse duration. The main result is concerned with frequency modulation. It is shown that for some well-known baseband encoding schemes, such as bipolar, pair selected ternary, high density bipolar, and bipolar withnzeros substitution, the spectrum exhibits lines no matter what the value of the modulation index.

Spectral analysis is performed for a digital message produced by a variable-length encoder driven by a stationary memoryless source. In particular, closed form expressions are derived for both the continuous and the discrete part of the spectral density. The continuous part turns out to be a rational function ofz=\exp (j2\pi fT),whereTis the symbol period, whereas the discrete part exhibits in general spectral lines at multiple integers of(\lambda_{0}T)^{-1}, where\lambda_{0}is the greatest common divisor of the codeword lengths. As an application of the theory explicit formulas are derived for the spectra of BnZS and HDBncodes.

Variable-length word sequences (VLWS) and symbol sequences obtained from VLWS are discussed. A VLWS is regarded as a vector-valued stochastic process in which the sequence of vector dimensions is itself a stochastic process. For a given VLWS, the symbol sequence is not uniquely defined because it depends on how the choice of the time origin is made. In particular, a deterministic choice leads to a nonstationary symbol sequence even if the originating VLWS is stationary. A suitable random choice must be made in order to get a stationary symbol sequence from a stationary VLWS. Attention is given to the spectral analysis of both the VLWS and the corresponding stationary symbol sequence. An application to a VLWS consisting of mutually independent words is given.