The BCJR algorithm is renowned for its optimal equalization, minimizing bit error rate (BER) over intersymbol interference (ISI) channels. However, its complexity grows exponentially with the channel memory, posing a significant computational burden. In contrast, the linear minimum mean square error (LMMSE) equalizer offers a notably simpler solution, albeit with reduced performance compared to the BCJR. Recently, Neural Network (NN) based equalizers have emerged as promising alternatives. Trained to map observations to the original transmitted symbols, these NNs demonstrate performance similar to the BCJR algorithm. However, they often entail a high number of learnable parameters, resulting in complexities comparable to or even larger than the BCJR. This paper explores the potential of NN-based equalization with a reduced number of learnable parameters and low complexity. We introduce a NN equalizer with complexity comparable to LMMSE, surpassing LMMSE performance and achieving a modest performance gap from the BCJR equalizer. A significant challenge with NNs featuring a limited parameter count is their susceptibility to converging to local minima, leading to suboptimal performance. To address this challenge, we propose a novel NN equalizer architecture with a unique initialization approach based on LMMSE. This innovative method effectively overcomes optimization challenges and enhances LMMSE performance, applicable both with and without turbo decoding.

An online shaping technique for high performance communication over Gaussian channels with Inter-Symbol Interference (ISI) and receiver Analog to Digital Converter (ADC) noise is presented. The technique uses online transmitter precoding over Pulse Amplitude Modulation (PAM) constellation, designed to shape the symbols distribution so that peak power constraint at the channel output is satisfied. An iterative decoder shares information between a modified M-BCJR module, which computes online the trellis transition probabilities of the shaped distribution, and turbo decoder. The result is a reduction in the required Effective Number Of Bits (ENOB) of the receiver ADC. We show that the transmitter precoding performs channel equalization without using any filter. At the limit of very high Signal to Noise Ratio (SNR) a complete channel inversion is possible so that the receiver signal spectrum becomes almost flat. Theoretical bounds are analytically derived which enable to assess the possible gain using the proposed method. For data rates of 200 Gbps and 400 Gbps over printed circuit board, simulations show that the shaping scheme enables reduction in the ENOB requirement as high as 1.43 bit and 1.78 bit, respectively, compared to uniform 4-PAM transmission with turbo equalization at the receiver side.

An online shaping technique for high performance communication over Gaussian channels with Inter-Symbol Interference (ISI) and receiver Analog to Digital Converter (ADC) noise is presented. The technique uses online transmitter pre-coding over Pulse Amplitude Modulation (PAM) constellation, designed to shape the symbols distribution so that peak power constraint at the channel output is satisfied. An iterative decoder shares information between a modified M-BCJR module, which computes online the trellis transition probabilities of the shaped distribution, and turbo decoder. The result is a reduction in the required ADC Effective Number Of Bits (ENOB), which is in particular attractive in modern high-speed wireline links. Theoretical bounds are analytically derived which enable to assess the possible gain using shaping. On practical scenarios aim to transmit 200 Gbps and 400 Gbps over printed circuit board, we demonstrate in simulations an overall ENOB gains as high as 1.43 bit and 1.78 bit, respectively, compared to uniform 4-PAM transmission with turbo equalization at the receiver side.

In this paper, we present a new low-complexity iterative joint phase estimation and decoding algorithm which can be applied to low-density-parity-check (or turbo) codes for channels affected by strong phase noise. The algorithm exhibits a very good performance and a very low complexity even in strong phase noise, high code rate, and high-order constellations. The proposed algorithm works by first obtaining a preliminary pilots-aided phase estimation. Next, the expressions for the belief propagation messages are approximated around that reference and reduced to a Gaussian canonical form. This results in a simple closed form of a recursive update. Simulations results for phase-shift-keying modulation of size 8 and quadrature-amplitude modulation of size 64 are presented and compared with other algorithms in the literature.

A novel PAPR shaping technique intended for systems with high spectral efficiency and single carrier transmission is presented. This technique applies puncturing of error correction redundancy bits to achieve a desired Markovian symbol transmission probability distribution, which avoids symbol sequences with high peak values. For a 16-QAM system with transmission rate of 3 bit/symbol and a root raised cosine (RRC) pulse shape filter with roll-off factor 0.1 and duration of six symbols, an overall gain of about 2.2 dB is demonstrated by simulations using pragmatic binary turbo code for error correction. The technique can be used to improve any single carrier communication link limited by the high power amplifier (HPA) peak power.

Phase noise poses a serious challenge for high-speed digital communications systems mainly when going to higher and higher carrier frequencies, such as in satellite communications. Traditionally, phase noise estimation was performed separately from the decoding task and it was shown, recently, that there is much to be gained from joint estimation and decoding, particularly when using LDPC (low-density parity check)/turbo codes. However, jointly estimating phase noise and decoding is a very complex and computationally demanding task. In this paper, we propose several algorithms based on the sum and product algorithm (SPA) for low complexity joint decoding and estimation of coded information in strong phase noise channels. These algorithms are based on a novel approximation of SPA messages as Tikhonov mixtures of a given order. Since mixture-based Bayesian inference such as SPA, creates an exponential increase in mixture order for consecutive messages, a mixture reduction scheme is a must. Therefore, in this paper, we propose a low complexity mixture reduction algorithm, which provably satisfies an upper bound on the Kullback Leibler (KL) divergence between the mixture and the reduced mixture. We then reduce the complexity even further, including limiting the model order and reducing the clustering effort to simple component selection. As an extreme case, it is even possible to reduce the number of modes to one. We show the relation between the simplified algorithm to the phase locked loop (PLL). Finally, we show simulation results and complexity analysis for the proposed algorithms, which show superior performance over other state of the art low complexity algorithms.

In this contribution, we propose a signal constellation for the the phase noise channel which does not require pilots thus increasing the effective information rate in a communication system. This constellation does not present rotational symmetry thus enabling decoding algorithms such as SPA, to converge without the use of pilots. We will provide Bit Error Rate (BER) simulations which show the superiority of this constellation over standard MPSK with pilots. Moreover, we will provide a method to analyze any arbitrary signal constellation and provide a figure of merit for its performance when iterative decoding algorithms are used.

This paper proposes an improved receiver for the Tomlinson-Harashima precoder (THP). In this paper we demonstrate that most of the capacity loss associated with conventional THP is due to the suboptimal receiver, which is a simple modulo operation. We then demonstrate how this loss can can be avoided with the optimization of the receiver. In the suboptimal receiver the modulo operation ignores the dependency between the implicitly added sequence of symbols removed by it. We then implement a THP with quantized outputs. The quantized outputs enable to implement a Maximum A-Posteriori Probability (MAP) receiver that exploits the dependency between the received symbols. The MAP receiver is named Enhanced MODulo operation (EMOD), and operates as an inner decoder within an iterative scheme, either with outer Low Density Parity Check (LDPC) code or outer Turbo Code. Simulation results show considerable gains relative to coded conventional THP, and also relative to the Turbo-Equalizer (TE) (which does not use a precoder in the transmitter).

This paper presents a method to find a lower bound on the peak power reduction of single carrier signals having low rolloff pulse shaping filter, and transmitted over the Additive White Gaussian Noise channel (AWGN). Peak power as well as any other property can be controlled by judiciously choosing a codebook, such that the peak power of any of the codewords does not exceed a threshold (or exceeds in a low probability). One of the methods to generate such codebook, called shaping, is to start with a larger codebook with i.i.d. uniform distribution and choose a subcode which optimizes the property. This approach includes the popular method of Trellis Shaping. We analyze the limit of the gains obtainable by the shaping method by using the conditional limit theorem under Markov conditioning, and by that we can show achievable peak reduction bounds. We analyze two types of receivers. The first is a receiver that is matched to the subcode, i.e., it is aware of the dilution made at the transmitter, and the second is a mismatched decoder, a decoder for the original large codebook, like in the case of Trellis Shaping. We show that significant peak and Peak to Average (PAR) gains, as large as 1.7 dB, are achievable, even for small constellations such as 16QAM and 16APSK transmission constellations for Square Root Raised Cosine filter (SRRC) with rolloff 0.1.

This work addresses the multiple input single output (MISO) channel when the transmitted waveform at the antenna belongs to small discrete constellation. We do not allow precoding matrix between the modulators and the antennas that modifies the constellation at the antenna. In the proposed scheme, the constellation at each antenna is the original modulation by limiting the available precoding to relative phase shift and relative gain between the antennas. This way, simple modulation hardware can be used, as well as high efficiency power amplifiers working near saturation. The mutual information of the MISO Gaussian channel with channel state information (CSI) at the receiver and the transmitter is maximized by simple beamforming, if the input constellation is dense. We show that in case the constellation at the antenna is small, like MPSK, the beamforming solution does not lead to the highest rate. The optimal solution is obtained by precoding and joint signal shaping through nonuniform constellation transmission. A gain is achieved since the combination of few small constellations provides richer constellation in the receiver than the original constellation. We discuss in detail the case of a system with two transmitting antennas and one receiving antenna and QPSK modulation. We address two types of power constraints: the Tx power constraint and Rx power constraint, wherein in each case the capacity is maximized at the respective power constraint. We show that the optimal solution is highly dependent on the SNR and on the type of constraint. For example in the case of Tx power constraint, beamforming is the optimal solution in low SNR while in medium to high SNR nontrivial solutions are obtained. Additionally, the shaping distribution behaves inversely to the typical shaping: instead of Maxwell-Boltzmann distribution, we obtain what we call Inverse Maxwell-Boltzmann distribution.

This paper presents a novel analytical expression which approximates the bit error rate (BER) of the joint phase and symbol maximum a posteriori (JMAP) sequence estimator for: trellis code modulation (TCM) with M-ary phase shift keying (MPSK) modulation; any arbitrary phase noise model (i.e., not limited to the Wiener process); and either a matched or mismatched decoder. First, we derived convenient closed-form expressions for approximating the pairwise error probability of two code sequences for both matched and mismatched decoders. Since the expressions are formulated either in the time or in the frequency domain, it is possible to indicate the contribution of every frequency in the phase noise spectrum. We then applied the union bound on the code sequence pairwise errors. The analytical expression was tight (usually <;1dB) for MPSK constellations with M ≥ 4 and code rates ≥ 0.5. Once developed, the expressions will assist designers to consider the influence on receiver performance of the code characteristics, decoder implementation and RF synthesizer phase noise. It further enables joint optimization of the RF synthesizer, the code and the decoder for achieving the lowest error rate or other design targets.

In this paper we investigate a new class of codes, the 2-D tail-biting convolutional codes (2-D TBCCs). The class of two-dimensional convolutional codes (2-D CCs) is a littleresearched subject in coding theory, and tail-biting versions were hardly mentioned, unless they can be represented as a product of two 1-D codes. These codes have interesting geometry since they are the 2-D analog of the 1-D TBCC which their graph is a ring. The result being a code invariant to shifts in 2-D space. We apply algebraic methods in order to find bijective encoders, create parity check matrices and inverse encoders. Next, we discuss minimum distance and weight distribution properties of these codes. We observe that some of these codes exhibit very good codes performance. We then present several novel iterative suboptimal algorithms for soft decoding 2-D CCs, which are based on belief propagation and generalized belief propagation. The results show that the suboptimal algorithms achieve respectable results, in some cases coming as close as 0.4dB from optimal (maximum-likelihood) decoding.

This letter presents a reduced complexity equalizer for offset quadrature phase-shift keying (OQPSK), Gaussian minimum shift keying (GMSK), or any other staggered modulation format. Due to the relative time offset of the in-phase and quadrature components, the received signal must be sampled at a rate of at least twice the symbol rate of its corresponding nonstaggered signal, where using a symbol-spaced equalizer (SSE) is an option. The conventional approach doubles the number of equalizer coefficients required relative to QPSK. In this work it is shown that also for OQPSK it is possible to implement an equalizer spaced in symbol rate, with about half the complexity required for the linear equalizer (LE). For the decision feedback equalizer (DFE), savings are achieved only in the feed forward filter.

We consider nonlinear n-dimensional modulations for transmission of uniformly distributed analog source. We show that under certain conditions, a class of constant stretch modulations yields purchase of n dB in resulting SNR for every dB of transmitted power while keeping the probability of threshold effect constant. We also derive the optimal mappings for a class of modulations with two-dimensional spiral trajectory and show that constant stretch modulations are close to optimal in this case. Receiver design for two-dimensional case is proposed.

As communication extends to higher carrier frequencies, the phase noise problem becomes more severe and conventional phase tracking methods become inadequate. Jointly maximum likelihood sequence estimation (JMLSE) phase tracking and decoding is a practical tracking method achieving near optimal performance. There is a lack of an analytical tool for performance evaluation of coded JMLSE. In this paper we will present a novel approximated union bound on the performance of the JMLSE receiver for trellis coded MPSK and Wiener phase noise. Our analysis leads to an equivalent model of the JMLSE process which gives important insights. Since the union bound requires the summation of infinite number of error event, we introduce an efficient algorithm for selecting the error events with the significant contribution. The developed tool is usually tight and efficient, hence can be instrumental in finding codes which achieve low error rate on high phase noise channels.

Sparse equalizers, in which only a small subset of the filter taps is selected to be nonzero, were recently proposed as a low-complexity solution for receivers operating in wireless frequency-selective channels with sparse power profiles. The performance of the sparse equalizer heavily depends on its tap-positioning algorithm. This paper presents efficient low-complexity algorithms for determination of sparse equalizer tap positions based on a forward sequential search. We develop low-complexity metrics for the evaluation of the candidate tap positions in the search space as well as methods to effectively reduce the search space size. The proposed algorithms are shown to be superior over previously proposed algorithms in a wide range of channel conditions. Actually, the proposed algorithms yield, in most of the tested cases, performance identical to the optimal, prohibitively complex, tap-positioning algorithm. The main emphasis is on linear equalization suitable for wideband code-division multiple-access systems but the algorithm can be extended to a variety of equalization schemes and channels.

In this paper, we present an algorithm for joint carrier phase estimation and turbo decoding for the case of rapidly varying carrier phase during the transmitted block. The proposed algorithm shows improved performance over previously proposed communication schemes, both coherent and noncoherent, for channels with additive white Gaussian noise and high carrier phase noise. The novel algorithm utilizes a modified "two dimensional" bit carrier phase a posteriori probability (BCAPP) decoder containing additional states representing the received carrier phase. The BCAPP decoder calculates two extrinsic metrics: one representing the bit soft value and the other representing the received carrier phase probability density function approximation. A modified structure of the turbo code iterations is suggested, implementing separate propagation of the two metrics between the BCAPP decoders. One additional attractive property of the suggested algorithm is its robustness against phase noise model mismatch.

In this paper, the combination of decision feedback equalizer and digital phase-locked loop (DPLL) is investigated in detail in channels impaired by intersymbol interference and phase noise. Various methods for carrier phase recovery in several system configurations are analyzed and compared. Two cases of phase noise are considered. In the first one, the significant phase noise is in the receiver side, whereas in the second case, the significant noise is in the transmitter side. We select the best estimation schemes for the two cases and analyze them in terms of residual phase jitter, optimal loop gain, and complexity of implementation. Their performances relative to the other schemes are verified by simulations under both high and low signal-to-noise ratio conditions, where in the latter, the mean time to lose lock criterion is used. We show the relevance of knowing the source of the phase noise. We also treat the special case of one-dimensional signal constellations, such as binary phase-shift keying and gaussian minimum-shift keying (GMSK), where further optimizations are employed. Finally, the more general case of phase noise, which originates both in the transmitter and the receiver in the same order of magnitude is introduced. For this case, the new transmitter phase noise loop is combined with the receiver loop to a double DPLL.

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We propose a simple decoder for a widely used array code, known as the EVENODD code, which is originally designed to correct phased burst errors, to make it useful for correcting nonphased errors. The proposed scheme is capable of correcting almost all bursts up to a certain length. We show that the failure rate is sufficiently small and approaches zero as the block length increases. The redundancy of the code is twice the maximal burst length, which is a lower bound for the redundancy of a true burst-error-correcting code. Both the encoder and the decoder have very low complexity, both in terms of number of operations and in terms of computer code size

We show that a widely used array code, known as the even-odd code, which is targeted at phased burst errors, may also be useful for non-phased burst errors. A new decoder is proposed for this code which effectively converts it into a more general burst error correcting code. The proposed scheme is shown to be capable of correcting almost all bursts up to a certain length, such that its performance is attractive for many communication applications. Since the failure rate is sufficiently low, the code can be practically classified as a burst error correcting code. The redundancy in this code is equal to twice the maximal burst length, which is the same redundancy as the lower bound of conventional burst error correcting codes (the Reiger bound). Both the encoder and the decoder have very low complexity, both in terms of number of operations and in terms: of computer code size. We analyze the probability of failure, provide tight upper and lower bounds, and show that asymptotically this probability approaches zero for large blocks.

We propose a new turbo-encoding scheme for high spectral efficiency with performance close to the Gaussian channel capacity. The scheme combines nonuniform signaling on a Gaussian channel with pragmatic turbo-coded modulation (TCM) for simple and flexible implementation. A variable-rate turbo code is followed by a Huffmann code mapping onto nonequiprobable points in a quadrature amplitude modulation constellation. The rate of the turbo code is matched to the Huffmann code by variable puncturing, such that both the input bit rate and the output symbol rate are constant. It is shown that the new scheme provides shaping gains of 0.6 and 0.9 dB, at rates 2 and 3 b/dimension, respectively, compared with the equiprobable pragmatic TCM, and reach about 1 dB from the continuous input Gaussian channel capacity.

The definition of geometrically uniform (GU) codes, which exhibit the property of uniform error probability (UEP), had led to the work of finding such codes for the coherent detection with additive white Gaussian noise (AWGN) channel scenario. The extension of this notion for noncoherent detection as UEP for wider channel and decoder condition were referred to as general uniform error (GUE) codes. We consider the problem of finding good GUE codes for noncoherent detection using the independent overlapped observation noncoherent maximum likelihood sequence estimator decoder, for L/spl times/ MPSK modulation transmitted over AWGN channels. We list good codes for noncoherent decoding. We checked both rotational variant and invariant codes and, due to complexity reduction in the invariant case, show their advantage for noncoherent decoding.

We propose a new turbo encoding scheme for high spectral efficiency with performance close to the Gaussian channel capacity. The scheme combines nonuniform signaling on a Gaussian channel, and pragmatic turbo coded modulation for simple and flexible implementation. A table is used to map equiprobable input symbols into nonequiprobable points in the QAM constellation. It is shown that the new scheme provides shaping gains of 0.6 dB and 0.93 dB, at rates 2 and 3 bits/dim respectively compared to the equiprobable pragmatic turbo coded modulation, and reach about 1 dB from the Gaussian channel capacity.

This paper investigates the subject of reducing the complexity of turbo equalization in which a receiver combines the equalization and decoding process in an iterative fashion. We show that it is possible to approximate the a posteriori probability (APP) module by two time varying linear transversal filters and a simple nonlinear memoryless processor, and thus replace the exponential complexity of the APP module with quadratic complexity in the filter length. Further complexity reduction leads to linear complexity, for which the transversal filters are constant during the block. For this structure, which is similar to a previously suggested scheme, we calculate the optimized parameters and increase performance. The derivations cover the cases of BPSK and QAM modulations. Simulation results are presented for parallel-concatenated turbo code with BPSK and QAM modulation over channels that introduce severe amplitude distortion.

We propose a new turbo code noncoherent detection scheme based on a novel a-priori probability processor (APP). Using this method, many turbo encoders can be used without a change in the code or in the modulation, and without a significant change in the structure of the iterative decoder. Simulation results of this method are presented for various phase noise levels. We show a degradation of 0.7 dB from the coherent detection of the same codes.

A new design method is presented for the design of PLL loop filters for carrier recovery, bit timing, or other synchronization loops given the phase noise spectrum and noise level. Unlike the conventional designs, our design incorporates a possible large decision delay and S-curve slope uncertainty. Large decision delays frequently exist in modern receivers due to, for example, a convolutional decoder or an equalizer. The new design also applies to coherent optical communications where delay in the loop limits the laser linewidth. We provide an easy-to-use complete design procedure for second-order loops. We also introduce a design procedure for higher order loops for near-optimal performance. We show that using the traditional second-order loop is suboptimal when there is a delay in the loop, and also shows large improvements, either in the amount of allowed delay, or the phase error variance in the presence of delay.

We propose a signal processing technique, based on the estimate-maximize algorithm, in order to perform multiuser code-division multiple-access (CDMA) detection. This algorithm iteratively seeks for the maximum-likelihood solution. The resulting structure is a successive interference cancellation scheme which can be applied to both synchronous and asynchronous CDMA. Higher performance than similar methods is obtained from using deterministic annealing and multiple stages. A soft output is defined, and the signal-to-noise ratio in the soft output of the detector is measured for predicting performance with an outer code with soft input decoder. The new receiver is applied to the problem whereby in a synchronous CDMA system the orthogonality of the codes is destroyed by a frequency-selective channel, caused by multipath fading. This nonlinear technique is shown to perform much better than the minimum mean-square-error linear solution and several other algorithms. The algorithm lends itself to an efficient DSP or VLSI implementation. We evaluate the performance by simulations with coherent quadrature phase-shift keying modulation, known channel and long random Rayleigh multipath. In most cases, we set the number of users equal to the processing gain for maximal throughput. The results are also presented in the form of outage probabilities for random Rayleigh multipath against required fading margin.

This paper presents a new application of a suboptimal trellis decoding algorithm for combined equalization and decoding. The proposed algorithm can outperform the reduced-state sequence estimator (RSSE) of the same order of complexity. The algorithm, termed estimated future decision-feedback algorithm (EFDFA), was originally proposed for the problem of noncoherent decoding with multiple-symbol overlapped observations and is now reformulated for the problem of intersymbol interference inflicted channels. The EFDFA uses the RSSE as a building block. The performance improvement is achieved by using estimated future symbols in the decision process. The estimated future symbols are obtained by RSSE decoding time-reversed blocks of the input. The same technique can be used to greatly enhance the performance of the conventional decision-feedback equalizer. An analysis of the performance of the EFDFA based on the performance of the RSSE is described. The EFDFA can be configured as an adaptive equalizer capable of operating in a time-varying environment, and is shown to perform well in fading conditions. With only minor additional complexity, the EFDFA is also capable of producing soft outputs.

A new design method is presented for the design of PLL loop filters for carrier recovery, bit timing or other synchronization loops given phase noise spectrum and noise level. Unlike the conventional designs, our design incorporates a possible large decision delay and S-curve slope uncertainty. Large decision delays frequently exists in modern receivers due to, for example, a convolutional decoder or an equalizer. The new design also applies to coherent optical communications where delay in the loop limits the laser line width. We provide an easy to use complete design procedure for second order loops. We also introduce a design procedure for higher order loops for near optimal performance. We show that using the traditional second order loop is suboptimal when there is a delay in the loop, and also show large improvements, either in the amount of allowed delay, or the phase error variance in the presence of delay.

This paper introduces an improved nonlinear parallel interference cancellation scheme for code-division multiple access (CDMA) that significantly reduces the degrading effect on the desired user of interference from the other users that share the channel. The implementation complexity of the scheme is linear in the number of users and operates on the fact that parallel processing simultaneously removes from each user a part of the interference produced by the remaining users accessing the channel the amount being proportional to their reliability. The parallel processing can be done in multiple stages. The proposed scheme uses tentative decision devices at the multiple stages to produce the most reliably estimated received data for generation and cancellation of user interference. Simulation results are given for a multitude of different situations, in particular, those cases for which the analysis is too complex.

We consider the problem of uniform error property (UEP) for a coded modulation with constant energy multidimensional symbols, transmitted over the additive white Gaussian noise (AWGN) or fading channels and received by a broad class of decoders. This class includes coherent, partially coherent, double differential, and noncoherent decoders, decoders designed for fading channels, decoders using one or multiple-symbol observations, and many more. These decoders are described as special cases of a general decoder model. This decoder operates by maximizing an arbitrary likelihood function that its arguments are front-end correlator (matched-filter) outputs, A group code structure that guarantees UEP is developed by using the theory of geometrically uniform codes and applying it to the general decoder. These codes are defined over groups (commonly nonbinary) with isometric mapping to channel symbols. We show the code construction for the specific case of Lth-dimensional M-ary phase-shift keying (MPSK). An additional interesting property of these general uniform error codes is related to the case of noncoherent decoding. We show that when using codes of this family, if a code is noncoherently catastrophic, then it is also rotationally invariant. Then, the use of preceding of the input such that the code becomes rotationally transparent will also make it noncatastrophic.

Trellis coded modulation with two or multidimensional signal constellations, together with coherent maximum-likelihood detection, is considered an attractive solution for communications over the additive white Gaussian noise (AWGN) channel. In this paper a new noncoherent communication system is introduced called noncoherent coded modulation (NCM) as an alternative to coherent coded modulation. NCM achieves almost the same power efficiency, without bandwidth expansion or an extensive increase in complexity. As a noncoherent system, the method does not need carrier phase estimation. Nonetheless, differential encoding is not required. High performance noncoherent detection is achieved by using multiple symbol observations. Unlike previous approaches, a sliding window for the observations is used, with each observation covering several branches of the trellis, such that the observations are time-overlapped. We define a new type of noncoherent maximum-likelihood sequence estimator (MLSE), and analyze its performance over the AWGN channel by numerical calculation of the union bound. We perform a computerized search and present new codes for noncoherent detection with their performance. The new codes cover many useful rates and complexities and achieve higher performance than existing codes for noncoherent detection. The method can also be used for multiple symbol demodulation of MDPSK with better results than existing methods.

Noncoherent decoding of trellis codes using multiple-symbol overlapped observations was shown previously to achieve close to the coherent performance. Optimal decoding by the Viterbi algorithm for L-symbol observations requires a number of states which grows exponentially with L. Two novel suboptimal algorithms are presented, for which the number of states is the same as the original code, yielding a complexity depending weakly on L. For practical values of L, both algorithms are substantially less complex than the optimal algorithm. The first algorithm, the basic decision feedback algorithm (BDFA), is a low complexity feedback decoding scheme, based on the Viterbi algorithm. This algorithm is shown to suffer from increased error probability and from error propagation. A slight modification to this algorithm can, in most cases, reduce these effects significantly. The second algorithm uses the BDFA as a basic building block. This algorithm is based on a novel concept called "estimated future" and its performance is very close to optimum for most practical eases with some additional complexity and memory requirements as compared to the first algorithm. Performance analysis and simulation results are also given.

A new series expansion is developed for the probability distribution function and the cumulative distribution function for indefinite noncentral Hermitian quadratic forms in complex normal random variables. The moment generating function is inverted by contour integration using the residue theorem. The function is separated into two parts, one part, containing an essential singularity, is expanded by Laurent series and the other part is expanded by Taylor series. The series are combined for evaluating the residue of the complete function. Several different series can be obtained by modifications of the basic approach. The series are computationally efficient and normally fast converging. The convergence rate depends on the separation of the eigenvalues. Multiple eigenvalues are allowed, and can be used to approximately replace a close pair of eigenvalues.

Practical M-FSK systems experience a combination of time and frequency offsets (errors). This paper assesses the deleterious effect of these offsets, first individually and then combined, on the average bit error probability performance of the system. Exact expressions for these various error probability performances are derived and evaluated numerically for system parameters of interest. Also presented are upper bounds on average symbol error probability for the case of frequency error alone which are useful in assessing the absolute and relative performance of the system. Both continuous and discontinuous phase M-FSK cases are considered when timing error is present, the latter being much less robust to this type of offset.>