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.

For a binary system specified by the density operators p0and p1and by the prior probabilities q0and q1Helstrom's theory permits the evaluation of the optimal measurement operators and of the corresponding maximum correct detection probability. The theory is based on the eigendecomposition (EID) of the operator, given by the difference of the weighted density operators, namely D = qlρl- q0ρ0. In general, this EID is obtained explicitly only with pure states, whereas with mixed states it must be carried out numerically. In this letter we show that the same evaluation can be performed on the basis of a modified version of the Gram matrix. The advantage is due to the fact that the outer products of density operators are replaced by inner product, with a considerable dimensionality reduction. At the limit, in quantum optical communications the density operators have infinite dimensions, whereas the inner products are simply scalar quantities. The Gram matrix approach permits the explicit (not numerical) evaluation of a binary system performance in cases not previously developed.

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 1986, Laurent showed that binary continuous phase modulated (CPM) signals can be decomposed into pulse amplitude modulated (PAM) waveforms. This result, later extended to multilevel CPM signals by Mengali and Morelli, has attracted much attention, especially for the design of simplified receivers. The paper proceeds from the premise that the decomposition into PAM waveforms is not unique, and follows a systemtheory approach that leads to a natural formulation. First, the CPM modulator is represented by a nonlinear part (encoder) followed by a linear part (bank of PAM modulators).The two parts are then rearranged with the aim of obtaining real-valued continuous PAM waveforms and to reduce their number. Several classes of decompositions arise with this approach, each of which admits an infinite number of different decompositions of the same CPM signal, among which the Laurent and MengaliMorelli results appear as special cases. Throughout the paper, a systematic use is made of the Kronecker product, which appears to be the natural tool for dealing with CPM decompositions and for expressing results in a very simple form.

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.

The well known decomposition of a CPM signal into PAM waveforms represents the linear part of a CPM modulator. The paper deals with the non-linear part, not fully developed in the literature. It is shown that this part can be represented by a finite-state synchronous sequential machine (SM). In such a way the CPM modulator can be implemented by a SM followed by bank of interpolating filters. Three different architectures are obtained using three PAM decompositions. This architecture can also be used to represent multi-hCPM. Finally, examples of application are developed.

This letter proposes a new pulse amplitude modulation decomposition of continuous phase modulation (CPM) signals under the constraint of phase response separability. The separability condition is met by a broad class of CPM signals, including full-response signals. We show that the proposed decomposition has minimal cardinality of N/sub c/=L(M-1), where M is the alphabet size and L is the CPM memory length. The cardinality increases linearly with L, while previous proposed decompositions have cardinalities M/sup L/-M/sup L-1/, which grow exponentially with L.

A continuous phase modulation (CPM) is specified by the size M of the input alphabet, the modulation index h, the phase response /spl phi/(t) and the duration LT of the active part of /spl phi/(t). The integer L indicates the memory and, roughly, the complexity of a CPM system. In this contribution we show that a CPM with an arbitrary memory L is equivalent to a CPM with a unitary memory and a bigger alphabet with size L(M - 1) + 1. The assumption is that the phase response /spl phi/(t) is "separable", a requirement verified in some practical implementations. We discuss some consequences of this equivalence on modulator and demodulator structure.

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.

This paper gives a closed-form evaluation of the spectrum of the pulse interval modulated (PIM) signal. In particular, we take into account the issue of "stationarization" (in the strict sense) of a PIM signal, which is a necessary task, preliminary to that of spectral analysis. This aspect was never considered in the literature and gives deeper insight into the nature of PIM signals. The solution to both these problems (stationarization and closed-form evaluation of the spectrum) lies upon the theory of variable-length vector-valued random processes.

The paper deals with an OFDM (orthogonal frequency division multiplexing) system based on filter-bank architecture. The known implementation uses a DFT (discrete Fourier transform) processor and a polyphase network (PPN). Even if it is based on complex components, in the final step it operates the real part extraction of the incoming signal. This leads to redundant operations in the DFT processor and in the PPN. Specifically, for the transmission of N complex symbol sequences at a given rate ½F, an N-point DFT processor and an N-branch PPN, both working at the rate F, are required. This implementation can be improved with a complexity reduction by a factor of two. In fact, in the paper an architecture is presented based on an N-point DFT processor and N-branch PPN at the rate F, for the transmission of 2N (in place of N) complex symbol sequences at the rate ½F

A discrete-time system described by a truncated Volterra series whose input is a Markov chain is considered. A general explicit formula is derived for the mean value of the output process in terms of the transition-probability matrix of the input and of the Volterra kernels.

A new scheme is derived for the digital TDM-FDM conversion (transmultiplexer) ofNvoice channels. In the complex version it simply consists of 1) a polyphase filter and 2) a modulator with carrierW2Nt2=exp(j2πt2/2N) wheretis the normalized discrete time, both working at the rate of the TDM (or FDM) flux. The polyphase filter is quite different from the polyphase network encountered in other transmultiplexer schemes: it is one-input one-output with a constant phase in each sideband of the resulting FDM signal. The modulator replaces the bankNmodulators or theN-point DFT processors of usual schemes. The derivation of the scheme is carried out in general, and its complexity is discussed with reference to the standard format ofN=64 voice channels.

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.

The second-order analysis of the output of a discrete-time nonlinear system described by a truncated Volterra series whose input consists of a sequence of independent random variables (white noise)is considered. The main result consists of an explicit formula for the mean value of the output process in terms of the cumulants of the input and of Volterra kernels. This formula, with suitable modifications, allows the calculation of the output correlation as well as of the continuous and discrete components of the spectral distribution. Several applications are considered, and in particular the Gaussian white noise case is worked out in detail. Finally, the computational aspects of the analysis are discussed with the aim of showing that in several situations closed-form results can be obtained.

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.

An approach to evaluate the error probability in conventional PAM digital data transmission systems with correlated symbols in the presence of intersymbol interference and additive noise is formulated in general and it is applied to the Gram-Charlier series expansion method. It is shown that the technique of conditioning few symbols before and/or after the symbol to be detected increases substantially the range of signal-to-noise ratios with an acceptable increase of numerical work. This technique also improves existing bounds on error probability, as, e.g, Glave's bound.

A general theory is presented for evaluating the spectral density of time jitter produced in a self-timed regenerative repeater. The timing circuit consists of a general memoryless nonlinear device followed by a narrow-band tank circuit tuned as close as possible to the symbol rate; the signal (timing wave) at the output of the circuit has both amplitude and phase modulations that cause the timing error (jitter). The theory includes the cases of arbitrary pattern statistics, pulse waveform, and nonlinear processing, in contrast to previous work which is strongly limited to the assumptions of pattern symbol independence and particular nonlinearities. The time jitter is treated as a discrete-parameter random process and its spectral density is finally related to the input message statistics, pulse waveform, and timing circuit parameters. In most cases the spectral density turns out to be a rational function of exp (j2\pifT),fbeing the frequency andTthe symbol period. Hence, a spectral factorization can be obtained that leads to a useful linear equivalent circuit of the timing (nonlinear) circuit (NLC). This last feature is illustrated by three examples.

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.

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.

Block codes are multilevel codes of constant length whose codewords are the channel encodings of binary sequences of constant length. The spectral density calculation of pulse trains resulting from the use of these codes is dealt with. Under the assumption that the encoding be represented by a finite-state synchronous sequential machine, by means of homogenous Markov chain theory, the spectral density is evaluated in closed form. Both the continuous and the discrete spectral components are easily obtained as soon as the encoder functions are specified. The result obtained is particularly attractive since the finite-state machine model can be used to represent a broad class of encoding schemes of engineering interest. As an example, the spectral density is calculated in the case of the Franaszek MS-43 code.

This paper deals with a decomposition of a set of stationary random processes:x_{1}, x_{2}, \cdots, x_{n}. The decomposition has the form:x_{1} = y_{11}, x_{2} = y_{21} + y_{22}, x_{3} = y_{31} + y_{32} + y_{33}, etc., where the componentsy_{ij}have the following properties: for a fixedi, they are completely correlated in pairs; for a fixedj, they are uncorrelated in pairs. Assuming the spectral matrix of thex_{i}'s as known, the spectral description of they_{ij}'s given by a lower triangular matrix, is determined. This is achieved by both an iterative and a direct method. In both methods regular and singular cases are considered.

This paper deals with the time jitter caused by noise in a trigger circuit; specifically with its probability density (PD). The model adopted for the circuit assumes that the initiation of switching occurs at the instant when the front of the input signal crosses some reference level. The time jitter is defined as the fluctuation of the initiation time due to the presence of an additive noise. It is shown that, when the corrupted signal has one crossing at the reference level, the PD of the jitter is given by the crossing rate at the same level. This rate, in turn, is known in the case of a general uncorrupted signal added to a stationary Gaussian noise. Hence, the PD is determined for various examples of uncorrupted signals. In many cases, the PI) tends rapidly to the Gaussian law as the signal-to-noise ratio (SNR) becomes larger. In other cases the deviation from the Gaussian law remains significant even for large SNRs. It is also shown that the one-crossing situation occurs in most cases of practical interest, specifically when the SNR is fairly large and the noise is spectrally of the same order of magnitude as the signal. Finally, the case is considered where the signal and the additive noise are prefiltered by low-pass systems.