In this paper, we derive a phase-noise estimator for MIMO systems affected by independent phase noise at each antenna. The proposed phase-noise estimator employs the expectation-maximization algorithm and can incorporate channel decoding. The main feature of the proposed estimator lies in the fact that it estimates separately phase-noise samples at the transmitter and at the receiver, instead of estimating, as it is typical in single-antenna case, the sum of the samples at the two sides. Simulation results show that the proposed estimator allows obtaining a very good performance at an affordable complexity.

The analysis of the channel capacity in the absence of prior channel knowledge (noncoherent channel) has gained increasing interest in recent years. Yet, the noncoherent channel capacity is still unknown for the general case. In this paper, we derive bounds on the capacity of the noncoherent, underspread complex Gaussian, orthogonal frequency division multiplexing, wide sense stationary channel with uncorrelated scattering, under a peak power constraint or a constraint on the second and fourth moments of the transmitted signal. These bounds are characterized only by the system signal-to-noise ratio (SNR) and by a newly defined quantity termed effective coherence time. Intuitively, the effective coherence time can be interpreted as the length of a block in the block-fading model in which a system with the same SNR will achieve the same capacity as in the analyzed channel. Unlike commonly used coherence time definitions, it is shown that the effective coherence time depends on the SNR, and is a nonincreasing function of it. We show that for low SNR, the capacity is proportional to the effective coherence time, while for higher SNR, the coherent channel capacity can be achieved provided that the effective coherence time is large enough.

In this paper we propose new capacity achieving modulation schemes for the peak-power limited Gaussian channel. Applying a multilayer approach to continuous phase modulations (CPM), we obtain schemes whose capacity approaches the capacity of the AWGN channel while exhibiting a low peak to average power ratio (PAPR). Moreover, thanks to the structure of CPM, the nonlinear characteristic of the RF power amplifier can be efficiently compensated with slight additional complexity. We compare multilayer CPM with APSK modulations and observe improvements both in capacity and PAPR.

In this paper we improve the spectral efficiency of continuous-phase modulations (CPM) when used in frequency-division multiplexing (FDM) systems with tight carrier frequency spacing. We show that significant improvements can be achieved applying a low-complexity iterative inter- channel interference (ICI) cancellation algorithm. Further improvements are obtained in coded CPM systems with joint ICI cancellation and iterative decoding.

The pragmatic approach to coded continuous-phase modulation (CPM) is proposed as a capacity-achieving low-complexity alternative to the serially concatenated CPM (SC-CPM) coding scheme. In this paper, we first perform a selection of the best spectrally efficient CPM modulations to be embedded into SC-CPM schemes. Then, we consider the pragmatic capacity (a.k.a. BICM capacity) of CPM modulations and optimize it through a careful design of the mapping between input bits and CPM waveforms. The so obtained schemes are cascaded with an outer serially concatenated convolutional code to form a pragmatic coded-modulation system. The resulting schemes exhibit performance very close to the CPM capacity without requiring iterations between the outer decoder and the CPM demodulator. As a result, the receiver exhibits reduced complexity and increased flexibility due to the separation of the demodulation and decoding functions.

This paper deals with LT and Raptor codes used on a noiseless channel, as it is the case for the BEC or the AWGN channel with high SNR. The performance measure for such family of codes is the efficiency, i.e., the average amount of redundancy needed to decode the information bits. We consider the new class of graph-based LT and Raptor codes, we analyze their efficiency and we show that the optimal degree distribution of the LT code depends on the rate of the precoder.

In this paper we propose a semi-random technique for the generation of a class of eIRA codes (a popular class of irregular LDPC codes that can be encoded in linear time) suited to partially parallel decoder implementations. The suggested technique tries to avoid both low-weight codewords and nearcodewords, which limit the performance of the belief propagation decoder. Its effectiveness is verified by comparison with literature results. The obtained codes are versatile, in terms of code-rate and block length, and they are characterized by a low error floor.

The paper extends the pragmatic approach to coded continuous-phase modulation (CPM). It first proposes an optimization of the CPM capacity for given complexity and bandwidth efficiency. Then, the best CPM schemes are embedded into a pragmatic coded-modulation scheme, whose pragmatic capacity (a.k.a. BICM capacity) is maximized through a careful design of the mapping between input bits and CPM waveforms. The so obtained CPM schemes show a pragmatic capacity very close to the CPM capacity. An outer binary serially-concatenated convolutional code, cascaded with the CPM modulator through an interleaver, yields performance close to the pragmatic (and so to the CPM) capacity without requiring iterations between outer code and CPM modulator.

This paper deals with precode-only (PCO) Raptor codes and turbo fountain codes used on a noiseless channel, as it is the case for the BEC or the AWGN channel with high SNR. The performance measure for such family of codes is the efficiency, i.e., the amount of redundancy needed to decode the information bits. Following a previous paper, we give an exact expression for the average efficiency of maximum-likelihood (ML) decoded Hamming codes and maximal-length codes. Moreover, we give an expression of the variance of the efficiency. Finally, we improve the bound on the efficiency of turbo fountain codes, which are iteratively decoded.

The paper describes a few simple solutions for a two-antenna receiver diversity combining scheme applied to the mobile terminal of a WCDMA cellular system. All schemes assume that combination takes place right after the low noise amplifiers following the receiving antennas, thus avoiding the need for duplication of the receiver chain. Combining strategies include optimal combining, suboptimal combining and selection combining, in addition to a new, very simple scheme exploiting the turbo code diversity. Different schemes are compared by simulating them in a very realistic 3GPP scenario, with a receiver including Rake, turbo decoder and channel estimation. To check the obtained performance with information-theoretical limits, a mixed analytical-simulation technique is used to evaluate bounds to the outage probability. The paper also addresses the problem of the feedback control algorithms that govern the RF combination through some suitable baseband signal processing.

In this paper, an innovative joint-source channel coding scheme is presented. The proposed approach enables iterative soft decoding of arithmetic codes by means of a soft-in soft- out decoder based on suboptimal search and pruning of a binary tree. An error-resilient arithmetic coder with a forbidden symbol is used in order to improve the performance of the joint source/channel scheme. The performance in the case of transmission across the AWGN channel is evaluated in terms of word error probability and compared to a traditional separated approach. The interleaver gain, the convergence property of the system, and the optimal source/channel rate allocation are investigated. Finally, the practical relevance of the proposed joint decoding approach is demonstrated within the JPEG 2000 coding standard. In particular, an iterative channel and JPEG 2000 decoder is designed and tested in the case of image transmission across the AWGN channel.

In this paper, the design of a fully analog iterative decoder for a serially concatenated convolutional code is presented. The decoder is reconfigurable in both block length and code rate. An interleaver size up to 2400 bit is considered. The decoder core implements a single SISO working on a window of the whole code trellis. It is then reused several times to decode the two constituent codes. The resulting decoder performs iterations, but it is fully analog. The extrinsic information exchanged in the decoding process is stored in an analog memory and permuted through a reconfigurable interleaver. Behavioral analysis of the decoder as well as precision and mismatch impact on performance are reported in the paper.

Although communication systems are often analyzed under the assumption of prior knowledge of the channel, this is rarely the case in practical systems, which usually need to estimate the channel from the received communication signal. The analysis of channel capacity in the absence of prior channel knowledge had gained increasing interest in recent years, but this capacity is not yet known for the general case. In this paper we derive tight bounds on the capacity of the channel in the case of slow frequency selective fading channels with peak power constraint. We also show that in this case, a system that employs truncated Gaussian signaling and minimum Euclidean distance receiver can achieve information rates which are very close to the channel capacity.

In this paper we propose a technique to implement in a parallel fashion a turbo decoder based on an arbitrary permutation, and to expand its interleaver in order to produce a family of prunable S-random interleavers suitable for parallel implementations. We show that the spread properties of the obtained interleavers are almost optimal and we prove by simulation that they are very competitive in terms of error floor performance. A few details on the decoder architecture are also provided.

In this paper we discuss the design of linear time encodable extended irregular repeat-accumulate (eIRA) codes for bit-interleaved coded modulation (BICM) systems. We interpret eIRA codes as a particular instance of the recently introduced multi-edge type (MET) low-density parity-check (LDPC) codes. This allows us to take into account during the design step both the systematic nature of these codes and the different level of protection given to each bit by the choice of input alphabet and mapping function. Furthermore, we extend to the eIRA-BICM case the method used by Roumy et al., IEEE Trans. Inf. Theory, 2004, to optimize LDPC codes for the binary input AWGN (bi-AWGN) channel. This method is based on EXIT functions and leads to a linear program. Using this method we design capacity-approaching eIRA codes for 4PAM and 8PSK. For both short and long block lengths these codes have performance approximately 0.1 dB worse than LDPC codes optimized with density evolution and with similar decoding complexity.

In this paper, we present an all-analog implementation of the rate-1/3, block length 40, UMTS turbo decoder. The prototype was designed and fabricated in a 0.35$mu$m CMOS technology and operates at 3.3 V. We also introduce a discrete-time first-order model for analog decoders which allows fast BER simulations, while taking into account circuit transient behavior and component mismatch. The model is applied to the rate-1/3 analog turbo decoder for UMTS defined in the 3GPP standard, and the discrete-time model predictions are compared with the decoder experimental performance and the transistor-level simulations. These results demonstrated that this model can be successfully used as a tool to both predict analog decoder performance and give design guidelines for complex decoders, for which circuit-level simulations are impractical.

In this paper, we present an all-analog implementation of the rate-1/3, block length 40, universal mobile telecommunications system (UMTS) turbo decoder. The prototype was designed and fabricated in 0.35$\mu$m complementary metal-oxide-semiconductor technology and operates at 3.3 V. We also introduce a discrete-time first-order model for analog decoders which allows fast bit-error rate simulations, while taking into account circuit transient behavior and component mismatch. The model is applied to the rate-1/3 analog turbo decoder for UMTS defined in the Third Generation Partnership Project standard, and the discrete-time model predictions are compared with the decoder experimental performance and the transistor-level simulations. These results demonstrated that this model can be successfully used as a tool to both predict analog decoder performance and give design guidelines for complex decoders, for which circuit-level simulations are impractical.

This correspondence presents an upper bound on the minimum distance of serially concatenated codes with interleaver where the inner code is a systematic recursive convolutional encoder and the outer code is any convolutional encoder. The resulting expression shows that the minimum distance of the concatenated code cannot grow more than O(K1-1/df(O)), where K is the information word length, and df(O)is the free distance of the outer code. The obtained upper bound is shown to agree with and, in some cases, improve over previously known results

In this paper, we discuss the design and testing results of an analog 0.35 /spl mu/m CMOS turbo decoder for the rate-1/3, 40 bit UMTS turbo code. The prototype was successfully tested at nominal conditions (2 Mbit/s), with an overall power consumption of 10.3 mW at 3.3 V. The tested BER curve shows a limited performance loss (about 0.5 dB) with respect to that of the digital implementation. We also discuss a discrete-time model of the analog decoder which allows us to run BER simulations including circuit transient behavior and device mismatch in a very short time. Circuit-level simulations demonstrate the validity of our model. According to the discrete-time simulation, a significant contribution to the performance loss is due to device mismatch.

In this paper we propose a semi-random technique to design irregular LDPC codes with low error floor. In particular, the suggested technique is related to a popular class of irregular LDPC codes that can be encoded in linear time, and tries to avoid both low-weight codewords and near-codewords, which limit the performance of the belief propagation decoder. Besides, an easy and effective rule to design puncturing patterns for this class of LDPC codes is provided. The effectiveness of the new design criteria is proved by comparison with literature results.

In a previous paper as presented by Tarable et al. (2004), the authors introduced the concept of mapping functions as a tool to cope with the problem of collision-free memory mappings in turbo and LDPC parallel decoder implementations. In this paper, we address some implementation issues of the original solution, define a dual problem, which can be of high interest for practical implementations, and give the generalization of mapping functions to N > 2 parallel decoders.

In this paper, we propose two techniques to design good S-random interleavers, to be used in parallel and serially concatenated codes with interleavers. The interleavers designed according to these algorithms can be shortened, in order to support different block lengths in such a way that all the permutations obtained by pruning, when employed in a parallel turbo decoder, are collision-free. The first technique, suitable for short and medium interleavers, guarantees the same performance of nonparallel interleavers in terms of spreading properties, simulated frame-error probabilities, and obtainable minimum distance of the actual codes. The second algorithm, to be used for large block lengths, permits achieving high degrees of parallelism at the price of a slight degradation of the spread properties, and also to change the degree of parallelism on-the-fly. The operations of a parallel turbo decoder employing these interleavers are described, and an example of the advantages of the proposed techniques is provided in a realistic system framework.

We deal with the design of interleavers in a coded code-division multiple-access (CDMA) scenario, where at the receiver an iterative turbo-like structure to perform multiuser detection is employed. The choice of the interleavers affects both the maximum-likelihood (ML) performance and the impact of the suboptimality of the iterative receiver. First, heuristic criteria of goodness for a set of interleavers, each assigned to a given active user, are introduced and motivated. One of these criteria is based on the intersection between the equivalent codes seen after the interleavers for each user pair. The design rules are valid for any kind of channel code. In particular, when the channel code used by every user is a terminated convolutional code, a very simple design rule, in the subset of congruential interleavers, is specified. The suitability of an interleaver set to iterative decoding is also treated. The analysis leads to a design rule which is shown to have great importance on the performance of a turbo-like receiver. Numerical results assess the validity of the derived design rules by showing that, for iterative multiuser receivers and reasonable block lengths, the suitability to iterative decoding is more important than the performance optimization.

In this paper, after a description of the principles of iterative (turbo) multiuser detection for CDMA systems, a new structure, based on a linear user separator, is introduced and analyzed. The new user separator processes the matched filter outputs according to a minimum mean-square-error (MMSE) criterion for the first few iterations, while it bypasses the MMSE filter when the interferers' bits are known with some reliability. This new receiver is tested by simulation and compared with other receivers previously introduced in literature. Also, a way of studying and designing iterative multiuser receivers is given by adapting the method of EXtrinsic Information Transfer (EXIT) charts. With the EXIT charts, the receiver performance (in an information-theoretic sense) can be evaluated as it evolves along the iterations. Thus, the receiver threshold can be reliably estimated.

In this paper, we propose two techniques to design good S-random interleavers, to be used in parallel and serially concatenated codes with interleavers. The interleavers designed according to these algorithms can be shortened in order to support different block lengths: all the permutations obtained by pruning, when employed in a parallel turbo decoder, are collision-free.

In this paper we present a comparison between coherent and noncoherent UWB receivers, under a realistic propagation environment, that takes into account also the effect of path-dependent pulse distortion. As far as coherent receivers are concerned, both maximal ratio combining (MRC) and equal gain combining (EGC) techniques are analyzed, considering a limited number of estimated paths. Furthermore, two noncoherent schemes are presented: a simple differential detector and its adaptive version. While the first receiver shows a significative performance loss, if compared to coherent detectors, the second one is able to guarantee probability of error similar to the one obtained employing a Rake receiver, equipped with up to 20 fingers, without requiring channel estimation.

This paper deals with the design of an optimum soft-input soft-output (SISO) a-posteriori receiver front-end for the downlink of direct-sequence W-CDMA cellular systems. The proposed structure yields the minimum bit error probability when used alone with hard decision at its output, and it evaluates the optimal metric (in the presence of the intersymbol interference (ISI) due to multipath propagation) to be provided to the turbo decoder. Performance of the new SISO equalizer are shown for both uncoded and coded systems in the standard framework of UMTS system.

In this paper, a concatenated coding scheme consisting of an outer algebraic block code performing hard decoding connected to an inner turbo code through a rectangular interleaver is analyzed. The performance of the concatenated code is evaluated through a semi-analytical approach, based on the expression of the word error probability at the output of the outer decoder as a function of the error statistics at the output of the turbo decoder estimated through simulation. The proposed technique yields results that depend on the size of the interleaver, and permits to obtain trade-offs between that size and the outer code characteristics.

The main problem of space-time trellis codes is constituted by their complexity, which grows exponentially with the number of transmit antennas. To avoid this shortcoming, one can think to a suboptimum decoder in which a simplified metric, together with a preliminary filtering step, is used. In this paper, we develop this idea and give some possible choices for the filter design. These different choices are compared with one another by means of analytical tools and simulations.

The design and test results of a three-metal, double-poly, 0.35 μm; CMOS analog turbo decoder for the rate-1/3, block length 40, UMTS turbo code, are presented. A discrete-time model of analog decoding networks is also presented. This model can be used as a tool to both predict chip performance in a short time and give design guidelines for complex decoders, for which circuit-level simulations are impractical.

A new upper bound on the minimum distance of turbo codes is derived. It improves over existing ones and shows that the minimum distance grows approximately with the base-3 logarithm of the information word length

This paper derives an upper bound on the minimum distance of serially concatenated convolutional codes. The resulting expression shows that their minimum distance cannot grow more than approximately K1-1/df(O), where K is the information word length, and df(O)is the free distance of the outer code

This correspondence deals with the design and decoding of high-rate convolutional codes. After proving that every (n,n-1) convolutional code can be reduced to a structure that concatenates a block encoder associated to the parallel edges with a convolutional encoder defining the trellis section, the results of an exhaustive search for the optimal (n,n-1) convolutional codes is presented through various tables of best high-rate codes. The search is also extended to find the "best" recursive systematic convolutional encoders to be used as component encoders of parallel concatenated "turbo" codes. A decoding algorithm working on the dual code is introduced (in both multiplicative and additive form), by showing that changing in a proper way the representation of the soft information passed between constituent decoders in the iterative decoding process, the soft-input soft-output (SISO) modules of the decoder based on the dual code become equal to those used for the original code. A new technique to terminate the code trellis that significantly reduces the rate loss induced by the addition of terminating bits is described. Finally, an inverse puncturing technique applied to the highest rate "mother" code to yield a sequence of almost optimal codes with decreasing rates is proposed. Simulation results applied to the case of parallel concatenated codes show the significant advantages of the newly found codes in terms of performance and decoding complexity.

In this paper, a new upper bound on the minimum distance of turbo codes is derived. The new bound is obtained by construction of an undirected graph which reflects the characteristics of the constituent codes and the interleaver. The resulting expression shows that the minimum distance of a turbo code grows approximately with the base-3 logarithm of the information word length. The new bound is easy to compute, applies to rate k/sub 0//n/sub 0/ constituent encoders, and often improves over existing results.

For high-data-rate applications, the implementation of iterative turbo-like decoders requires the use of parallel architectures posing some collision-free constraints to the reading/writing process from/into the memory. This consideration applies to the two main classes of turbo-like codes, i.e., turbo codes and low-density parity-check (LDPC) codes. Contrary to the literature belief, we prove in this paper that there is no need for an ad hoc code design to meet the parallelism requirement, because, for any code and any choice of the scheduling of the reading/writing operations, there is a suitable mapping of the variables in the memory that grants a collision-free access. The proof is constructive, i.e., it gives an algorithm that obtains the desired collision-free mapping. The algorithm is applied to two simple examples, one for turbo codes and one for LDPC codes, to illustrate how the algorithm works.

Recently, we proposed a new design technique to construct high-rate convolutional codes based on a structure formed by a block encoder and a simpler convolutional encoder (Graell i Amat, A. et al., IEEE Commun.. Lett., vol.5, no.11, p.453-5, 2001). The search technique was based on the optimization of the output weight enumerating function of the code. We now prove that every (n,n-1) convolutional code can be reduced to this structure. Following this result and suitably modifying our earlier search algorithm, we have been able to obtain the best (n, n-1) convolutional encoders to be used in the design of turbo codes. In this case, the search is aimed at the optimization of the input-output weight enumerating function of the encoders. We also derive an inverse puncturing method that can be applied to these high-rate convolutional codes to obtain a sequence of the (almost) best convolutional encoders. With such a method, a whole family of good encoders with different rates is obtained using the same encoder-decoder, thus permitting a great versatility that can be exploited in practical implementations.

A general method for the evaluation of the symbol error probability (SER) of both M-PAM and M-PPM UWB systems, in presence of multipath channel, multiuser and strong narrowband interference is presented. This method is shown to be able to include all the principal multiaccess techniques proposed so far for UWB, like time hopping and direct sequence. A comparison between the performance of three of these techniques is also presented, for both ideal RAKE receiver and MMSE equalizer, in an indoor communications scenario.

Ultra wideband system performance is analyzed in terms of error probability for different modulation and multi-access schemes over the AWGN channel. The analysis leads to closed-form expressions of the bit error probability under the hypothesis of chip-synchronous systems, allowing a comparison of different techniques in a possible scenario for UWB applications. The use of convolutional codes to improve the performance of time hopping systems is also addressed.

A new hard iterative decoding algorithm to decode the serial concatenation of two block codes separated by an interleaver is presented. The new scheme is based on the use of erasures as a sort of extrinsic information to be exchanged between the two decoders. Extensive simulations highlight the coding gain of the new algorithm with respect to standard hard iterative decoding.

We consider the application of concatenated codes with interleaver and iterative decoding to optical communication systems. We show how to obtain the optimum log-likelihood ratio to be provided to the soft decoder in the optical channel environment, and compare the performance of a decoder using it with the one employing a Gaussian approximation of the optical channel. Simulation results refer to practical turbo-product codes, and encompass the effect of quantization on the log-likelihood ratio. The results show that the Gaussian assumption in computing the log-likelihood ratio for the optical channel leads to significant losses.

Optimized high-rate convolutional codes are considered as the outer encoder of a serially concatenated structure where the inner encoder is replaced by the magnetic recording channel. Simulation results of the iterative decoding algorithm for an equalized Lorentzian channel model and a more realistic model that includes data-dependent transition noise are presented. The effect of precoder on performance is also studied, and simulation results are supported by EXIT chart analysis. All results refer to a comparison of the optimized codes with previously proposed schemes employing punctured codes or non optimized unpunctured codes with tail-biting decoding. Both trellis termination and tail-biting termination of the high-rate codes are studied. To terminate the code trellis we use the method derived by Amat, Montorsi and Benedetto, which only requires /spl nu/ (the code memory) tail-biting bits. Simulation results confirm the ML analysis: owing to their better distance properties, the scheme based on the new codes outperform state-of-the-art magnetic recording schemes based on both punctured and non optimized high-rate codes. The cost of using an unpunctured code versus the punctured one in terms of increased decoding complexity is turned into an advantage by applying to the high-rate code the soft-input soft-output (SISO) algorithm working on its dual trellis.

This paper considers the use of the best high-rate k/(k+1) convolutional codes obtained using the new construction technique described by Graell i Amat, Montorsi and Benedetto (see IEEE Communications Letters, vol.5, no.11, p.453-55, 2001) in a concatenated scheme. Simulation results for an AWGN channel and for a realistic magnetic recording channel are reported. It is shown that these codes, endowed with a decoding algorithm working on the dual code, yield performance improvements over the best known high-rate punctured codes with the same rate and memory in terms of both bit error probability and computational decoding complexity. For both the AWGN channel and the magnetic recording channel the new codes significantly lower the error floor with respect to known turbo-like code structures.

We address several aspects of high-rate convolutional codes. Some results from an exhaustive search for codes optimized with respect to their distance spectrum, and encoders optimized with respect to their input-output weight enumerating function are presented. An additive version of the dual-SISO algorithm suitable to decode such codes with limited complexity is described, together with simulation results for both stand-alone and concatenated codes showing the codes performance improvement.

The well-known uniform error property for signal constellations and codes is extended to encompass information bits. We introduce a class of binary labelings for signal constellations, called bit geometrically uniform (BGU) labelings, for which the uniform bit error property holds, i.e., the bit error probability does not depend on the transmitted signal. Strong connections between the symmetries of constellations and binary Hamming spaces are involved. For block-coded modulation (BCM) and trellis-coded modulation (TCM) Euclidean-space codes, BGU encoders are introduced and studied. The properties of BGU encoders prove quite useful for the analysis and design of codes aimed at minimizing the bit, rather than symbol, error probability. Applications to the analysis and the design of serially concatenated trellis codes are presented, together with a case study which realizes a spectral efficiency of 2 b/s/Hz.

We compare the convergence thresholds of parallel and serially concatenated codes with interleaver that employ iterative decoding algorithms working at bit or symbol level. The analysis technique is an extension to that of the symbol-level decoder presented by S. ten Brink (see Electron. Lett., vol.35, no.10, p.806-8, 1999), and is valid for very large interleavers. The comparison shows that working at symbol level always yields a lower convergence threshold. In the serial concatenation, the advantage is quite significant.

We present a thorough comparison among three alternative solutions for the transmission of medium-high speed data over mobile telecommunication systems with high quality of service. They are based on concatenated codes with interleavers, the first two using the parallel concatenation known as turbo code, and the third the serial concatenation recently proposed. The comparison between the two alternatives has been based on the computation of the free distance of the resulting codes, and on extensive simulations over additive Gaussian noise (AWGN) and correlated multipath fading channels.

Turbo codes have excellent performance at low and medium signal-to-noise ratios (SNR) very close to the Shannon limit, and are at the basis of their success. However, a turbo code performance curve can change its slope at high SNR if the code free distance is small. This "error floor" phenomenon is not acceptable for applications requiring very low values of bit error rates. A knowledge of the free distance and its multiplicity allows one to analytically estimate the error floor. An algorithm for computing the turbo code free distance, based on the notion of constrained subcodes, is described. Some considerations on the free distance distribution of turbo codes with growing interleaver length are also provided.

A new suboptimal turbo receiver for coded CDMA is presented. The user-separating front-end of this receiver has been thought of as the extension of linear multiuser detectors to an iterative structure. The complexity of this receiver is shown to be O(K), where K is the number of users. Simulations show that, for sufficiently high signal-to-noise ratios, the linear-US receiver tends to single-user performance.

We present a new algorithm for computing the free distance d/sub free/ of parallel and serially concatenated codes with interleavers, the parameter that dominates the code performance at very high signal-to-noise ratios (SNRs). The knowledge of d/sub free/ allows one to analytically estimate the error floor, which may prevent the use of concatenated codes in applications requiring very low error rates. The algorithm is based on the new notion of constrained subcodes, and permits the computation of large distances for large interleavers without a constraint on the input sequence weight (e.g., up to d/sub free/=40 for a rate-1/3 turbo code with interleaver length N=3568). Applications to practical cases of relevant interest, i.e., (1) the new Consultative Committee for Space Data Systems (CCSDS) standard for deep-space telemetry and (2) the new UMTS/3GPP standard for third-generation personal communications, are presented for the first time. Other related aspects, like a study on the free distance distribution of turbo codes with small/medium interleaver length, and a comparison between parallel and serial concatenation behavior, are also discussed.

We discuss the effects of quantization on the performance of the iterative decoding algorithm of concatenated codes with interleavers. Quantization refers here to the log-likelihood ratios coming from the soft demodulator and to the extrinsic information passed from one stage of the decoder to the next. We discuss the cases of a single soft-input soft-output (SISO) module, in its additive log-likelihood version (L-SISO), performing sequentially all iterations (an implementation solution coping with medium-low data rate as compared with the hardware clock), and that of a pipelined structure in which a dedicated hardware is in charge of each SISO operation (an implementation suitable for high data rates). We give design rules in both cases, and show that a suitable rescaling of the extrinsic information yields almost ideal performance with the same number of bits (five) representing both log-likelihood ratios and extrinsic information at any decoder stage.

We present in this paper a novel accurate method to analyze the performance of an optical link where the amplified spontaneous emission noise, enhanced by a fiber nonlinear phenomenon called parametric gain, is the limiting factor. Our method allows us to compute the exact error probability given a generic noise spectral density at the input of a direct detection optical receiver, using arbitrary optical and electrical filters. We compare our results with those predicted using the standard Gaussian technique (based on the Q factor), showing that this approximation may lead to significant errors. Our method is then used to evaluate the impact of parametric gain on a realistic long-haul multiwavelength link operating at 10 Gb/s, showing both the system limitation imposed by this phenomenon and the inaccuracy of the Q factor method.

In this paper, the basic theory of interleavers is revisited in a semi-tutorial manner, and extended to encompass noncausal interleavers. The parameters that characterize the interleaver behavior (like delay, latency, and period) are clearly defined. The input-output interleaver code is introduced and its complexity studied. Connections among various interleaver parameters are explored. The classes of convolutional and block interleavers are considered, and their practical implementation discussed. The trellis complexity of turbo codes is tied to the complexity of the constituent interleaver. A procedure of complexity reduction by coordinate permutation is also presented, together with some examples of its application.

Implementation rules for iterative decoders of concatenated codes with interleavers are proposed based on a study of the quantization effects on the performance. We consider both cases of a single soft-input soft-output (SISO) module performing sequentially all iterations and of a pipelined structure in which a dedicated hardware is in charge of each SISO operation. In the last case, we show that a suitable rescaling of the extrinsic informations yields almost ideal performance with the same number of bits (5) representing both LLRs and extrinsic information at any decoder stage.

A novel method to evaluate the bit error rate (BER) in optical links is described. It analyzes systems limited by amplified spontaneous emission (ASE) noise, enhanced by the non-linear parametric gain (PG) effect, and allows one to compute the exact error probability, given a generic optical noise spectral density at the input of a direct detection optical receiver. The method has been employed to evaluate the impact of PG on a 10 Gb/s per channel long-haul multi-wavelength system. It has also been used to show how the system performance evaluation based on the Q-factor Gaussian approximation may lead to erroneous BER evaluations.

We study the effect of parametric gain in long-haul, optically amplified links. We started by introducing a formalism required to evaluate the spectral enhancement of amplified spontaneous emission noise due to the Kerr-effect non-linear interaction between the signal and noise. These results are then used to evaluate parametric gain-induced performance degradation on the system Q value. Finally we analyze long-haul links at 10 Gb/s using different dispersion compensation maps by evaluating the maximum reachable distance in each configuration. We show that less frequent dispersion compensation improves the system performance.

A double serially concatenated code with two interleavers consists of the cascade of an outer encoder, an interleaver permuting the outer codeword bits, a middle encoder, another interleaver permuting the middle codeword bits, and an inner encoder whose input words are the permuted middle codewords. The construction can be generalized to h cascaded encoders separated by h-1 interleavers, where h>3. We obtain upper bounds to the average maximum likelihood bit-error probability of double serially concatenated block and convolutional coding schemes. Then, we derive design guidelines for the outer, middle, and inner codes that maximize the interleaver gain and the asymptotic slope of the error probability curves. Finally, we propose a low-complexity iterative decoding algorithm. Comparisons with parallel concatenated convolutional codes, known as "turbo codes", and with the proposed serially concatenated convolutional codes are also presented, showing that in some cases, the new schemes offer better performance.

Recursive systematic convolutional encoders have been shown to play a crucial role in the design of turbo codes. We recall some properties of binary convolutional encoders and apply them to a search for good constituent convolutional codes of turbo codes. Tables of the "best" recursive systematic convolutional encoders found are presented for various rates, together with the average bit-error probability performances of some turbo codes using them.

A serially concatenated code with interleaver consists of the cascade of an outer encoder, an interleaver permuting the outer codewords bits, and an inner encoder whose input words are the permuted outer codewords. The construction can be generalized to h cascaded encoders separated by h-1 interleavers. We obtain upper bounds to the average maximum-likelihood bit error probability of serially concatenated block and convolutional coding schemes. Then, we derive design guidelines for the outer and inner encoders that maximize the interleaver gain and the asymptotic slope of the error probability curves. Finally, we propose a new, low-complexity iterative decoding algorithm. Throughout the paper, extensive comparisons with parallel concatenated convolutional codes known as "turbo codes" are performed, showing that the new scheme can offer superior performance.

A serially concatenated code with interleaver consists of the cascade of an outer encoder, an interleaver permuting the outer codeword bits, and an inner encoder whose input words are the permuted outer codewords. In this paper we derive design guidelines for the outer and inner codes that maximize the interleaver gain and the asymptotic slope of the error probability curves.

The fast-paced evolution of long-haul and high-bit-rate terrestrial and submarine optical transmission links requires powerful analysis tools that take into account all the relevant phenomena in the fiber. To provide such a tool, we developed a time-domain optical system simulation package, integrated in the TOPSIM simulation environment. The fiber simulation module makes use of the vector form of the propagation equations to account for the quasi-degenerate two-mode (the two polarizations) medium propagation characteristics. This way, all polarization-related effects and their interplay with the other linear and nonlinear phenomena in the fiber can be accurately modeled. In particular, the fiber third-order susceptivity, responsible for all major nonlinear effects, is expressed in its actual vector form, so that nonlinear polarization mode coupling could be accounted for. Conventional birefringence and PMD are generated using appropriate random models. A novel feature of the simulator is that it uses time-domain digital filters to simulate dispersion effects, as opposed to the usual FFT-based algorithms. This approach leads to more efficient computing for a wide range of bandwidth and dispersion values. We present the fiber simulation module in detail. As an example of the use of the simulation package, the analysis of a long-haul two-channel transoceanic WDM transmission system is presented.

For the first time, a solution to the problem of recovering and tracking the signal constellation at the receiver side for binary and octonary digital optical transmission systems employing the modulation of polarization (POLSK) is proposed and analyzed. The effect of two recovery algorithms on the system performance is evaluated in terms of power penalty induced at P(e)=10/sup -9/, acquisition and tracking speed, and hardware complexity. The results show that very small penalties are obtainable in practical situations by properly designing the recovery algorithms.

A new decoding algorithm for geometrically uniform trellis codes is presented. The group structure of the codes is exploited in order to improve the decoding process. Analytical bounds to the algorithm performance and to its computational complexity are derived. The algorithm complexity does not depend on the number of states of the trellis describing the code. Extensive simulations yield results on the algorithm performance and complexity, and permit a comparison with the Viterbi algorithm and the sequential Fano algorithm.

A parallel concatenated convolutional coding scheme consists of two constituent systematic: convolutional encoders linked by an interleaver. The information bits at the input of the first encoder are scrambled by the interleaver before entering the second encoder. The codewords of the parallel concatenated code consist of the information bits followed by the parity check bits of both encoders. Parallel concatenated codes (turbo codes), decoded through an iterative decoding algorithm of relatively low complexity, have been shown to yield remarkable coding gains close to theoretical limits. We characterize the separate contributions that the interleaver length and constituent codes give to the overall performance of the parallel concatenated code, and present some guidelines for the optimal design of the constituent convolutional codes.

For pt.I see ibid., vol.42, no.3, p.751-65 (1996). In Part I, general results on rotationally invariant codes and encoders were derived assuming no algebraic structure. In Part II, trellis codes based on group systems are considered as a special case for which code and encoder constructions are particularly simple. Rotational invariance is expressed as an algebraic constraint on a group code, and algebraic constructions are found for both "absorbed precoder" encoders and for encoders with separate differential precoders. Finally, the various encoder forms used to achieve rotational invariance are compared based on their performance on an AWGN channel.

A parallel concatenated coding scheme consists of two simple constituent systematic encoders linked by an interleaver. The input bits to the first encoder are scrambled by the interleaver before entering the second encoder. The codeword of the parallel concatenated code consists of the input bits to the first encoder followed by the parity check bits of both encoders. This construction can be generalized to any number of constituent codes. Parallel concatenated schemes employing two convolutional codes as constituent codes, in connection with an iterative decoding algorithm of complexity comparable to that of the constituent codes, have been previously shown to yield remarkable coding gains close to theoretical limits. They have been named, and are known as, "turbo codes". We propose a method to evaluate an upper bound to the bit error probability of a parallel concatenated coding scheme averaged over all interleavers of a given length. The analytical bounding technique is then used to shed some light on some crucial questions, which have been floating around in the communications community since the proposal of turbo codes.

We present a theoretical framework for rotational invariance of trellis codes. The distinction between codes and encoders plays a pivotal role. Necessary and sufficient conditions for rotational invariance are derived under general assumptions, and a construction is presented that obtains a rotationally invariant encoder for almost any rotationally invariant code, independent of the code's algebraic structure. Encoders that use a differential precoder are considered as a separate case, where a system-theoretic characterization of precoding is used to find two alternative and slightly less general encoder constructions.

Receiver structures for the direct detection of binary and multilevel digital optical modulation schemes employing the modulation of the state of polarization of light, or polarization shift keying (POLSK), are proposed and accurately analyzed, in the presence of optical amplifier ASE noise and electrical receiver noise. A comprehensive set of results shows that the performance of the binary system is approximately 3 dB better than IM/DD (on peak optical power), and stays so for a wide range of optical filter bandwidths. As a somewhat unexpected result, the multilevel schemes show a lower sensitivity to the use of a wide optical filter than the binary one. As a consequence, transmitting 3 b/symbol on a cubic constellation on the Poincare sphere brings about virtually no penalty, whereas transmitting 2 b/symbol has a better performance than binary transmission, for even relatively low values of the optical filter bandwidth. A clear explanation of this phenomenon has been found. These results suggest that narrow-bandwidth and therefore low-dispersion, multilevel, POLSK transmission could be performed with no penalty. Together with the low excitation of nonlinear effects in the fiber that polarization modulation seems to ensure, these results make POLSK schemes look promising for very-long-haul transmission.>

New geometrically uniform trellis codes over multidimensional unbalanced 4-, 8-, and 16-PSK constellations, obtained with the group code approach, are presented. They improve by up to 1.25 dB the asymptotic performance of the best known trellis codes over multidimensional balanced PSK. To check their "optimality", an upper bound on the free distance of group trellis codes is derived. Some of the new codes achieve the largest obtainable free distance, and several others are close to the bound. Exploiting the symmetry properties of the codes, curves of tight upper bounds to the error probability are also presented.

The performance of optical coherent modulation schemes using the state of polarization of light as the modulating parameter (POLSK, POLarization Shift Keying) is analyzed, in the presence of laser phase noise. First, we compare three different analysis methods to account for the effects of phase noise in the binary case, and choose the one offering the best trade-off in terms of accuracy/complexity. Successively, we derive a tight upper bound to the performance of multilevel POLSK systems with postdetection filtering. Then, we include phase noise using the best method as tested in the binary case and extensively discuss the performance of binary and various multilevel POLSK constellations, optimized with respect to the IF filter bandwidth. The results show that binary POLSK has a very low sensitivity to phase noise and that multilevel POLSK is even less sensitive than the binary scheme.>

The application of the well-known technique of trellis coded modulation to coherent optical communications using polarization shift keying (POLSK) is described and analyzed. The resulting receiver is formed by a front-end which performs the heterodyne detection and the Stokes parameter extraction, cascaded with an electronic Viterbi processor operating the maximum likelihood estimate of the transmitted sequence. Results in terms of the error event probability using optimum as well as a simpler suboptimum branch metric show power gains of the order of 3-4 dB, at the expense of a reasonable increase in complexity, only concerning the processing in the electronic domain. These coding gains are not lost even in the presence of high levels of phase noise, to which POLSK in general is highly insensitive.>

The theory of group codes simplifies the construction of good rotationally invariant trellis codes for the additive white Gaussian noise channel. New geometrically uniform trellis codes, with spectral efficiency 3 bits/2D, based on multidimensional L/spl times/16-PSK constellations in 4, 6, and 8 dimensions, obtained by group trellis codes over nonbinary groups of the kind Z/sub 16//sup L/, are presented. Most of the codes improve, in terms of asymptotic gain and rotational invariance, the best codes known so far in the literature.>

The theory of group codes has been shown to be a useful starting point for the construction of good geometrically uniform codes. In this paper we study the problem of building multilevel group codes, i.e., codes obtained combining separate coding at different levels in such a way that the resulting code is a group code. A construction leading to multilevel group codes for semi-direct and direct products is illustrated. The codes that can be obtained in this way are identified. New geometrically uniform Euclidean-space codes obtained from multilevel codes over abelian and nonabelian groups are presented.>

A description of the algorithms to evaluate the main parameters determining the performance of a trellis-coded modulation (TCM) scheme is presented. TCM schemes are divided into classes that have an increasing degree of symmetry, so as to properly match the various algorithms to each class. The algorithms are compared in terms of computational complexity and tested on a set of multidimensional PSK codes.>

This paper, together with a companion one [l], constitute a thorough presentation and analysis of a series of techniques grouped under the denomination “Polarization Spreading,” the intent of which is that of making the performance of a coherent optical receiver insensitive to the polarization transformations occurring along the optical fiber, without resorting to polarization compensation or control. The principle on which Polarization Spreading operates is that of spreading the signal power over different states of polarization, so that heterodyning of at least half of the optical incident bit energy is always ensured. The main obtained results concern (presented in [l]) a condition that guarantees the maximum heterodyned signal energy, some classes of spreading waveforms satisfying the above condition, the derivation of the structure and performance of optimum receivers in Gaussian noise; and, presented in this paper, the analysis of a suboptimum but easily implementable receiver and a detailed comparative analysis of the spectral broadening induced by the different spreading techniques. The results encompass most of the binary optical coherent modulation schemes proposed and experimented, such as ASK, FSK and DPSK.

Multilevel digital coherent optical modulation schemes based on the state of polarization of a fully polarized lightwave are proposed and analyzed. Based on the complete statistical characterization of the Stokes parameters, extracted through appropriate signal processing in the presence of shot and additive Gaussian noise, the optimum maximum likelihood receiver operating symbol by symbol is derived. The exact performance in terms of the average symbol error probability is found. Optimum constellations for the case of equipower 4, 8, 16 and 32 signals are found on the basis of the minimization of the error probability for a given average power. Their performance turns out to be promising as compared to other standard modulation techniques. The spectral analysis of polarization modulated signals is presented. A new receiver structure, which solves the problem of the excess penalties incurred in the presence of channel dichroism, is proposed and analyzed.>

Presents an analysis of a series of techniques grouped under the denomination "polarization spreading", the intent of which is that of making the performance of a coherent optical receiver insensitive to the polarization transformations occurring along the optical fiber, without resorting to polarization compensation or control. The principle on which Polarization spreading operates is that of spreading the signal power over different states of polarization, so that heterodyning of at least half of the optical incident bit energy is always ensured. The main results obtained concern a condition that guarantees the maximum heterodyned signal energy, some classes of spreading waveforms satisfying the above condition, the derivation of the structure and the performance of optimum receivers in Gaussian noise. The results encompass most of the binary optical coherent modulation schemes proposed and experimented, such as ASK, FSK and DPSK.>

The theory of geometrically uniform signal sets and codes over groups is applied to the case of L/spl times/MPSK constellations. Conditions for rotational invariance of group codes are discussed. The tables of geometrically uniform partitions found in Benedetto et al. (1993) are used to construct good geometrically uniform trellis codes over nonbinary Abelian groups. The present authors consider L/spl times/4PSK and L/spl times/8PSK constellations used to transmit information rates of 1 and 2 bit/two dimensions, respectively; and present tables of good codes over generating groups (Z4)/sup L/ and (Z8)/sup L/,for L ranging from 1 to 4. In most cases, they improve the tables of codes known so far. Moreover, the geometrical uniformity of codes allows a very easy performance evaluation, so that the authors also present a complete set of curves of error event probability for the obtained codes.>

The theory of geometrically uniform trellis codes is applied to the case of multidimensional PSK (phase shift keying) constellations. The symmetry group of an L*MPSK (M-ary PSK) constellation is completely characterized. Conditions for rotational invariance of geometrically uniform partitions of a signal constellation are given. Through suitable algorithms, geometrically uniform partitions of L*MPSK (M=4,8,16 and L=1,2,3,4) constellations are found, which present good characteristics in terms of the set of distances at a given partition level, the maximum obtainable rotational invariance, and the isomorphism of the quotient group associated with the partition. These partitions are used as starting points in a search for good geometrically uniform trellis codes based on binary convolutional codes.>

A rigorous analysis of digital coherent optical modulation schemes using the state of polarization as the modulating parameter is presented. The analysis obtains the exact performance of all the polarization-based modulation schemes proposed in the literature so far, including a differential demodulation scheme, named DPOLSK, which does not require either electrooptic or electronic polarization tracking. Preliminary results involving multilevel transmission schemes based on the state of polarization are introduced. A spectral analysis of POLSK signals is also proposed.>

The authors present the results of a design study of the receiver in a digital transmission system using the combined coding and modulation schemes known as Ungerboeck codes. Specifically, they examine the design of the receiver for encoded 16-PSK (phase shift keying) modulation, presenting first the traditional structure for the optimum receiver and then a simpler structure. The decoding depth of the Viterbi algorithm, the quantization of the metrics inside the Viterbi processor, and the phase jitter in the recovered carrier are considered. The impact of branch and path metric quantization inside the receiver is discussed, showing that a reasonable number of bits (8) is sufficient to obtain nearly optimum performance when the code complexity is limited. The effect of imperfect carrier recovery inside the receiver is studied, providing accurate analytical estimates of the error event probability as well as an upper bound to the symbol error probability. Results of a detailed simulation, including carrier and bit timing recovery blocks, show that the effects of imperfections on the bit error probability are very small, even at low signal-to-noise ratios. On the whole, results show the robustness of the Viterbi algorithm with respect to fairly rough quantizations of the metrics and indicate that carrier recovery is not as critical as expected.>

The theoretical aspects of the encoding process are investigated, resulting in a precise definition of linear codes together with theorems that clarify how they can be obtained. A particular subset of linear codes, called superlinear codes, for which the performance analysis is highly simplified is identified. The most relevant performance measures for the analysis of this class of codes are discussed. The minimum Euclidean distance and the event and bit error probabilities are found analytically using the uniform error property (when applicable) or variations on it. This yields accurate upper and lower bounds to the error rate at the price of reasonable computational complexity. The theory is then applied to the search for 'good' codes and to their performance evaluation. The cases of 16- and 32-PSK codes, which are good candidates for use in digital satellite transmission, are considered. Several new results in terms of error event and bit error probabilities are presented, showing considerable gains in terms of SNR with respect to the uncoded case.>

The performance of combined coding and modulation schemes in channels with intersymbol interference and nonlinearities is studied using an analytical approach. The class of possible receivers is presented, and a generalization of the minimum Euclidean distance used for the asymptotic analysis of the performance of combined coding and modulation schemes on AWGN channels is introduced. Numerical results are presented for coded 16-PSK modulation.

TOPSIM III is a flexible, easy to use, stand-alone Fortran package for the digital simulation of communication systems in the time domain. TOPSIM III provides a simple high-level language for the description of the system. Programs can be written following the same structure used in drawing the system block diagram. The paper discusses the time domain simulation approach used in the design of the package and describes its capabilities. The contents of the subprogram libraries are presented, describing some of the building blocks that can be used in the simulation as well as the performance evaluation facilities available in the package. Some simple examples illustrate the use of TOPSIM III as a tool for analysis and design of digital communication systems.

Conventional techniques attempting to compensate for the influence of linear and nonlinear distortion in digital satellite channels include optimization of channel filtering. With conventional linear filtering it has been experienced that there is still room for considerable improvement, so that it appears reasonable to investigate some forms of nonlinear filtering in an attempt to cope with those distortions. In this paper a nonlinear equalizer structure is proposed, and its performance is analyzed. It is shown, by some examples Of application to a 4-phase PSK satellite channel, that it can prove quite effective in terms of a tradeoff between performance improvement and hardware complexity.

A good deal of effort has been spent, in the past few years, to devise numerical algorithms for evaluating the performance of digital communication systems over noisy linear channels, i.e., in the presence of intersymbol interference and noise. In this concise paper we present a method for computing the error probability of multilevel baseband digital modulation systems when the channel is nonlinear with memory. The algorithm is based on a Volterra series expansion of the nonlinearity; with this model, we show that the moments of the disturbance can be computed recursively, and the same techniques in use for linear channels can be applied for evaluating the error probability. This approach can be generalized to consider noise entering the nonlinear channel. The computing algorithms are described in detail, and a complete example is worked out.

Two main classes of receivers for data modems using linear modulation systems over time-dispersive channels have been investigated by many authors for both theoretical and practical purposes, 1) structure-constrained linear receivers, such as zeroforcing and mean-square-error tapped-delay-line equalizers, and 2) nonlinear receivers, such as decision-feedback equalizers and maximum likelihood sequence estimators. In this paper a linear receiver that turns out to be practical and optimum in the mean-square sense is analyzed in detail, and some interesting features of this receiver are stressed; for instance, it is shown that in the absence of noise it becomes a zero-forcing equalizer, provided that stability can be achieved. A comprehensive set of results is also presented, showing that conventional tapped-delay-line equalizers perform very close to the optimum.

An expression for the error probability of multilevel coherent phase-shift-keyed (CPSK) systems in the presence of intersymbol, interchannel, and co-channel interferences and additive Gaussian noise is derived. An exact expression is given for the binary CPSK system, whereas upper and lower bounds are presented for the multilevel systems. The approach proposed in this paper overcomes the difficulties of the exhaustive methods and allows accurate and fast evaluation of the character error probability. Only the impulse responses of the overall channels are needed, and the computational labor is reduced to the numerical evaluation of one particular type of integral, whose computation is based upon nonclassical Gaussian quadrature rules. The model of the system is general enough to allow the choice of a rectangular or otherwise-shaped modulating pulse and a constant or shaped envelope of the modulating carrier. Phase incoherence between adjacent carriers is also assumed, and a random misalignment among the modulating bit streams can be taken into account, if necessary. Extensive numerical results are presented for 2- and 4-level systems. The presentation of the results stresses a possible utilization of them in system design problems.

A new method is presented to compute the average probability of error in the presence of intersymbol interference and additive noise for multilevel pulse-amplitude-modulation (PAM) and partial-response-coded (PRC) signaling schemes. The method is based upon nonclassical Gauss quadrature rules (GQR) and suffers no limitation on noise statistics, so that it applies also for non-Gaussian noise. Moreover it yields some remarkable advantages as compared with other methods, in particular, with the series expansion method that has recently received considerable attention. Expressions for the truncation error are also given and their derivation is reported in the Appendix. Finally, examples of applications are presented and comparisons with other methods are carried out.

In multilevel bandpass data transmission there is usually a difference between the phase of the carrier signal and the phase of the receiver oscillator, thereby causing imperfect demodulation. The use of nonclassical Gauss quadrature rules (GQR's) allows: 1) a theoretical study of the joint effects of phase jitter, thermal noise, and intersymbol interference on average error probability; and 2) a maximum-precision sampling technique in the simulation of digital communication systems. On this basis, the mean-square-error and zero-forcing equalizers are considered, and their performances evaluated in terms of the average error probability for multilevel pulse-amplitude-modulation (PAM) and partial-response-coded (PRC) signaling schemes.

A computer simulation technique is used to evaluate the performance of the MAP strategy for sequential block orthogonal signaling over a Gaussian channel with noiseless and delayless feedback. The system performance is given in terms of the character error probability. The obtained results are compared with those derived using other suboptimum strategies previously proposed.